:: Algebraic Numbers :: by Yasushige Watase environ vocabularies NUMBERS, FINSEQ_1, SUBSET_1, FUNCT_1, ARYTM_3, TARSKI, NAT_1, XBOOLE_0, SUPINF_2, ZFMISC_1, GROUP_1, STRUCT_0, POLYNOM1, POLYNOM2, C0SP1, REALSET1, ARYTM_1, RELAT_1, CARD_1, XXREAL_0, VECTSP_1, ALGSTR_0, FUNCT_7, AFINSQ_1, CARD_3, MESFUNC1, POLYNOM3, FUNCSDOM, ORDINAL4, INT_2, GAUSSINT, BINOP_2, COMPLFLD, EC_PF_1, XCMPLX_0, FINSEQ_2, INT_3, ALGSEQ_1, POLYNOM4, PARTFUN1, IDEAL_1, CARD_FIL, VECTSP_2, POLYNOM5, HURWITZ, RATFUNC1, MSSUBFAM, QUOFIELD, BINOP_1, RING_1, RING_2, LATTICES, GCD_1, RLVECT_1, ALGNUM_1, RAT_1, INT_1, WAYBEL_8; notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, XTUPLE_0, MCART_1, DOMAIN_1, ORDINAL1, RELAT_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, NUMBERS, INT_1, FUNCOP_1, BINOP_1, FUNCT_4, FUNCT_7, SETWISEO, FINSEQ_1, FINSEQ_2, XCMPLX_0, XXREAL_0, CARD_1, XREAL_0, PBOOLE, VALUED_0, NAT_1, NAT_D, RAT_1, NEWTON, BINOP_2, MEMBERED, STRUCT_0, ALGSTR_0, C0SP1, NORMSP_1, VFUNCT_1, VECTSP_2, ALGSEQ_1, ALGSTR_1, RLVECT_1, GROUP_1, VECTSP_1, RINGCAT1, RING_3, GROUP_6, RING_2, POLYNOM1, UPROOTS, HURWITZ, RATFUNC1, BINOM, INT_3, GCD_1, RVSUM_1, COMPLFLD, POLYNOM3, POLYNOM4, POLYNOM5, FVSUM_1, PRE_POLY, REALSET1, COMPLEX1, EC_PF_1, GAUSSINT, IDEAL_1, RING_1, MSSUBFAM, QUOFIELD, RING_4; constructors TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, FUNCT_1, RELSET_1, NUMBERS, ORDINAL1, FUNCT_2, FINSEQ_1, FINSEQOP, STRUCT_0, ALGSTR_0, C0SP1, VECTSP_1, NORMSP_1, VFUNCT_1, POLYNOM3, SETWISEO, BINOP_1, REAL_1, RFINSEQ, FINSOP_1, BINARITH, VECTSP_2, GRCAT_1, REALSET2, QUOFIELD, ALGSTR_1, POLYNOM4, POLYNOM5, NAT_D, FVSUM_1, ALGSEQ_1, FUNCT_7, BINOP_2, INT_1, RLVECT_1, GROUP_1, RINGCAT1, MOD_4, GROUP_6, RING_3, RING_2, GROUP_4, FINSEQ_4, MATRIX_1, POLYNOM1, UPROOTS, HURWITZ, POLYNOM2, RING_4, XXREAL_2, XTUPLE_0, PARTFUN1, RVSUM_1, XCMPLX_0, RAT_1, NEWTON, VALUED_0, MEMBERED, NAT_1, FINSEQ_2, RATFUNC1, BINOM, GCD_1, IDEAL_1, EC_PF_1, GAUSSINT, AFINSQ_2, FUNCT_4, REALSET1, MSSUBFAM; registrations ALGSTR_0, GAUSSINT, CARD_1, INT_3, FINSEQ_2, FUNCT_2, INT_1, MEMBERED, NAT_1, NUMBERS, ORDINAL1, POLYNOM3, POLYNOM5, REALSET1, RELAT_1, RING_2, STRUCT_0, VECTSP_1, XREAL_0, RATFUNC1, RAT_1, XXREAL_0, COMPLFLD, XCMPLX_0, VALUED_0, POLYNOM4, ALGSTR_1, RLVECT_1, RING_1, RING_4, XBOOLE_0, RINGCAT1, RELSET_1; requirements NUMERALS, BOOLE, SUBSET, ARITHM, REAL; definitions VECTSP_1, GROUP_1, GROUP_6, ALGSTR_0; equalities BINOP_1, ALGSTR_0, REALSET1, FINSEQ_1, FINSEQ_2, XCMPLX_0, POLYNOM3, POLYNOM5, RATFUNC1, HURWITZ, STRUCT_0, GAUSSINT, INT_3; expansions STRUCT_0, SUBSET_1, FINSEQ_1, TARSKI, ALGSEQ_1, FUNCT_2, IDEAL_1, FUNCT_1, RING_2, ALGSTR_0, UPROOTS; theorems TARSKI, ZFMISC_1, FUNCT_1, XREAL_0, VECTSP_1, NORMSP_1, FINSEQ_1, FINSEQ_2, FINSEQ_3, NAT_1, XREAL_1, ORDINAL1, C0SP1, CARD_1, RELAT_1, XXREAL_0, SUBSET_1, GROUP_1, MATRIX_3, FVSUM_1, FUNCOP_1, POLYNOM3, POLYNOM5, RLVECT_1, IDEAL_1, BINOP_2, RING_3, GAUSSINT, INT_1, ALGSEQ_1, POLYNOM4, POLYNOM2, PARTFUN1, RING_2, RING_1, VECTSP_2, RING_4, XCMPLX_0, FUNCT_2, COMPLFLD, ALGSTR_0, STRUCT_0, GCD_1, EC_PF_1, RAT_1; schemes NAT_1, FINSEQ_2, FUNCT_2; begin :: Preliminaries reserve i,j for Nat; reserve A,B for Ring; theorem Th1: for L1,L2,L3 be Ring st L1 is Subring of L2 & L2 is Subring of L3 holds L1 is Subring of L3 proof let L1,L2,L3 be Ring; assume that A1: L1 is Subring of L2 and A2: L2 is Subring of L3; A3: the carrier of L1 c= the carrier of L2 & the addF of L1 = (the addF of L2) || the carrier of L1 & the multF of L1 = (the multF of L2) || the carrier of L1 & 1.L1 = 1.L2& 0.L1 = 0.L2 by A1,C0SP1:def 3; A4: the carrier of L2 c= the carrier of L3 & the addF of L2= (the addF of L3) || the carrier of L2 & the multF of L2= (the multF of L3) || the carrier of L2 & 1.L2= 1.L3 & 0.L2= 0.L3 by A2,C0SP1:def 3; set A1 = [:the carrier of L1, the carrier of L1:]; set A2 = [:the carrier of L2, the carrier of L2:]; set A3 = [:the carrier of L3, the carrier of L3:]; A8: the carrier of L1 c= the carrier of L3 by A3,A4; A9: the addF of L1 = (the addF of L2) || the carrier of L1 by A1,C0SP1:def 3 .= ((the addF of L3) || the carrier of L2) || the carrier of L1 by A2,C0SP1:def 3 .= (the addF of L3) || the carrier of L1 by A3,ZFMISC_1:96,RELAT_1:74; A10: the multF of L1 = (the multF of L2) || the carrier of L1 by A1,C0SP1:def 3 .= ((the multF of L3) || the carrier of L2) || the carrier of L1 by A2,C0SP1:def 3 .= (the multF of L3) || the carrier of L1 by A3,ZFMISC_1:96,RELAT_1:74; A11: 1.L1 = 1.L2 by A1,C0SP1:def 3 .=1.L3 by A2,C0SP1:def 3; 0.L1 = 0.L2 by A1,C0SP1:def 3 .=0.L3 by A2,C0SP1:def 3; hence thesis by A8,A9,A10,A11,C0SP1:def 3; end; theorem Lm1: F_Rat is Subfield of F_Complex by EC_PF_1:5,GAUSSINT:14,RING_3:48; theorem Th3: F_Rat is Subring of F_Complex by RING_3:43,Lm1; theorem Th4: INT.Ring is Subring of F_Complex by RING_3:47,Th3,Th1; Lm5: A is Subring of B implies In(0.A, B) = 0.B & In(1.A,B) = 1.B proof assume A1: A is Subring of B; then A2: In(0.A,B) = In(0.B,B) by C0SP1:def 3 .= 0.B by SUBSET_1:def 8; In(1.A,B) = In(1.B,B) by A1,C0SP1:def 3 .= 1.B by SUBSET_1:def 8; hence thesis by A2; end; Lm6: for a be Element of A st A is Subring of B holds a is Element of B proof let a be Element of A; assume A is Subring of B; then the carrier of A c= the carrier of B by C0SP1:def 3; hence thesis; end; Lm7: In(0.F_Rat,F_Complex) = 0.F_Complex & In(1.F_Rat,F_Complex) = 1.F_Complex & In(0.INT.Ring,F_Complex) = 0.F_Complex & In(1.INT.Ring,F_Complex) = 1.F_Complex by Lm5,Th3,Th4; theorem Th8: for x, y be Element of B, x1, y1 be Element of A st A is Subring of B & x = x1 & y = y1 holds x + y = x1 + y1 proof let x, y be Element of B, x1, y1 be Element of A; assume A is Subring of B; then the addF of A = (the addF of B) || the carrier of A by C0SP1:def 3; hence thesis by FUNCT_1:49,ZFMISC_1:87; end; theorem Th9: for x, y be Element of B, x1, y1 be Element of A st A is Subring of B & x = x1 & y = y1 holds x * y = x1 * y1 proof let x, y be Element of B, x1, y1 be Element of A; assume A is Subring of B; then the multF of A = (the multF of B) || the carrier of A by C0SP1:def 3; hence thesis by FUNCT_1:49,ZFMISC_1:87; end; registration let c be Complex; reduce In(c,F_Complex) to c; reducibility proof c in COMPLEX by XCMPLX_0:def 2; then c is Element of F_Complex by COMPLFLD:def 1; hence thesis by SUBSET_1:def 8; end; end; begin :: Define Extended eval Function for commutative rings A c= B :: based upon POLYNOM4 definition let A,B be Ring; let p be Polynomial of A; let x be Element of B; func Ext_eval(p,x) -> Element of B means :Def1: ex F be FinSequence of B st it = Sum F & len F = len p & for n be Element of NAT st n in dom F holds F.n = In(p.(n-'1),B) * (power B).(x,n-'1); existence proof deffunc G(Nat) = In(p.($1-'1),B)*(power B).(x,$1-'1); consider F be FinSequence of B such that A1: len F = len p and A2: for n be Nat st n in dom F holds F.n = G(n) from FINSEQ_2:sch 1; take y = Sum F; take F; thus y = Sum F & len F = len p by A1; let n be Element of NAT; assume n in dom F; hence thesis by A2; end; uniqueness proof let y1,y2 be Element of B; given F1 be FinSequence of B such that A3: y1 = Sum F1 and A4: len F1 = len p and A5: for n be Element of NAT st n in dom F1 holds F1.n = In(p.(n-'1),B)*(power B).(x,n-'1); given F2 be FinSequence of B such that A6: y2 = Sum F2 and A7: len F2 = len p and A8: for n be Element of NAT st n in dom F2 holds F2.n = In(p.(n-'1),B)*(power B).(x,n-'1); A9: dom F1 = Seg len p by A4,FINSEQ_1:def 3; now let n be Nat; assume A10: n in dom F1; then A11: n in dom F2 by A7,A9,FINSEQ_1:def 3; thus F1.n = In(p.(n-'1),B) *(power B).(x,n-'1) by A5,A10 .= F2.n by A8,A11; end; hence thesis by A3,A4,A6,A7,FINSEQ_2:9; end; end; theorem Th11: for n being Element of NAT, A,B be Ring, z be Element of A st A is Subring of B holds (power B).(In(z,B),n) = In((power A).(z,n),B) proof let n be Element of NAT, A,B be Ring, z be Element of A; assume A0: A is Subring of B; then z is Element of B by Lm6; then A2: In(z,B) = z by SUBSET_1:def 8; A3: 1_A = 1_B by A0,C0SP1:def 3; reconsider x = z as Element of B by A0,Lm6; (power B).(In(z,B),n) = In((power A).(z,n),B) proof defpred P[Nat] means (power B).(In(z,B),$1) = In((power A).(z,$1),B); A5: P[0] proof A6: In(1.A,B) = 1.B by A0,Lm5; In((power A).(z,0),B) = 1.B by A3,GROUP_1:def 7,A6; hence thesis by A3,GROUP_1:def 7; end; A7: for m be Nat st P[m] holds P[m+1] proof let m be Nat; assume A9: P[m]; A10: (power A).(z,m) is Element of B by A0,Lm6; A11: In((power A).(z,m),B) = (power A).(z,m) by A10,SUBSET_1:def 8; A12: (power A).(z,m+1) is Element of B by A0,Lm6; (power B).(In(z,B),m+1) = (power B).(In(z,B),m)* In(z,B) by GROUP_1:def 7 .= (power A).(z,m)*z by A0,A11,A2,Th9,A9 .= (power A).(z,m+1) by GROUP_1:def 7 .= In((power A).(z,m+1),B) by A12,SUBSET_1:def 8; hence thesis; end; for m be Nat holds P[m] from NAT_1:sch 2(A5,A7); hence thesis; end; hence thesis; end; theorem Th12: for x1,x2 be Element of A st A is Subring of B holds In(x1,B) + In(x2,B) = In(x1+x2,B) proof let x1,x2 be Element of A; assume A0: A is Subring of B; then x1 is Element of B by Lm6; then A2: In(x1,B) = x1 by SUBSET_1:def 8; x2 is Element of B by A0,Lm6; then A4: In(x2,B) = x2 by SUBSET_1:def 8; x1 + x2 is Element of B by A0,Lm6; then In(x1+x2,B) = x1+x2 by SUBSET_1:def 8 .= In(x1,B)+In(x2,B) by A0,A2,A4,Th8; hence thesis; end; theorem Th13: for x1,x2 be Element of A st A is Subring of B holds In(x1,B) * In(x2,B) = In(x1*x2,B) proof let x1,x2 be Element of A; assume A0: A is Subring of B; then x1 is Element of B by Lm6; then A2: In(x1,B) = x1 by SUBSET_1:def 8; x2 is Element of B by A0,Lm6; then A4: In(x2,B) = x2 by SUBSET_1:def 8; x1*x2 is Element of B by A0,Lm6; then In(x1*x2,B) = x1*x2 by SUBSET_1:def 8 .= In(x1,B)*In(x2,B) by A0,A2,A4,Th9; hence thesis; end; theorem Th14: for F be FinSequence of A, G be FinSequence of B st A is Subring of B & F = G holds In(Sum F,B) = Sum G proof let F be FinSequence of A, G be FinSequence of B; assume A0: A is Subring of B; defpred P[Nat] means for F being FinSequence of A, G being FinSequence of B st len F = $1 & F = G holds In(Sum F,B) = Sum G; P1: P[0] proof let F be FinSequence of A, G be FinSequence of B; assume A1: len F = 0 & F = G; then A2: F = <*>the carrier of A; A3: G = <*>the carrier of B by A1; In(Sum F,B) = In(0.A,B) by A2,RLVECT_1:43 .= In(0.B,B) by A0,C0SP1:def 3 .= 0.B by SUBSET_1:def 8 .= Sum G by A3,RLVECT_1:43; hence thesis; end; P2: for n being Nat st P[n] holds P[n+1] proof let n be Nat; assume A4: P[n]; let F be FinSequence of A, G be FinSequence of B; assume A5: len F = n+1 & F = G; reconsider F0 = F| n as FinSequence of A; n+1 in Seg (n+1) by FINSEQ_1:4; then A6: n+1 in dom F by A5,FINSEQ_1:def 3; rng F c= the carrier of A; then reconsider af = F.(n+1) as Element of A by A6,FUNCT_1:3; A7: len F0 = n by FINSEQ_1:59,A5,NAT_1:11; A8: len F = (len F0) + 1 by A5,FINSEQ_1:59,NAT_1:11; A9: F0 = F | dom F0 by A7,FINSEQ_1:def 3; reconsider G0 = G| n as FinSequence of B; n+1 in Seg (n+1) by FINSEQ_1:4; then A10: n+1 in dom G by A5,FINSEQ_1:def 3; rng G c= the carrier of B; then reconsider bf = G.(n+1) as Element of B by A10,FUNCT_1:3; A11: len F0 = n & F0 = G0 by FINSEQ_1:59,A5,NAT_1:11; G = G0^<*bf*> by A5,FINSEQ_3:55; then Sum G = Sum G0 + bf by FVSUM_1:71 .= In(Sum F0,B)+ bf by A4,A11 .= In(Sum F0,B) + In(af,B) by A5,SUBSET_1:def 8 .= In(Sum F0 + af,B) by A0,Th12 .= In(Sum F,B) by A5,A8,A9,RLVECT_1:38; hence thesis; end; for n being Nat holds P[n] from NAT_1:sch 2(P1,P2); hence thesis; end; theorem Th15: for n be Nat, x be Element of A, p be Polynomial of A st A is Subring of B holds In(p.(n-'1),B)*(power B).(In(x,B),n-'1) = In(p.(n-'1) * (power A).(x,n-'1),B) proof let n be Nat,x be Element of A, p be Polynomial of A; assume A0: A is Subring of B; then In(p.(n-'1) * (power A).(x,n-'1),B) = In(p.(n-'1),B)*In((power A).(x,n-'1),B) by Th13 .= In(p.(n-'1),B)*(power B).(In(x,B),n -'1) by A0,Th11; hence thesis; end; theorem Th16: for x be Element of A, p be Polynomial of A st A is Subring of B holds Ext_eval(p,In(x,B)) = In(eval(p,x),B) proof let x be Element of A, p be Polynomial of A; assume A0: A is Subring of B; consider F1 be FinSequence of B such that A1: Ext_eval(p,In(x,B)) = Sum F1 and A2: len F1 = len p and A3: for n be Element of NAT st n in dom F1 holds F1.n = In(p.(n-'1),B) * (power B).(In(x,B),n-'1) by Def1; consider F2 be FinSequence of A such that A4: eval(p,x) = Sum F2 and A5: len F2 = len p and A6: for n be Element of NAT st n in dom F2 holds F2.n = p.(n-'1) * (power A).(x,n-'1) by POLYNOM4:def 2; F1 = F2 proof A11: rng F2 c= the carrier of A; A8: dom F1 = dom F2 by A2,A5,FINSEQ_3:29; for k be Nat st k in dom F1 holds F1.k = F2.k proof let k be Nat; assume A10: k in dom F1; then F2.k is Element of A by A8,FUNCT_1:3,A11; then A13: F2.k is Element of B by A0,Lm6; F1.k = In(p.(k-'1),B) * (power B).(In(x,B),k-'1) by A3,A10 .= In(p.(k-'1) * (power A).(x,k-'1),B) by A0,Th15 .= In(F2.k,B) by A6,A10,A8 .= F2.k by A13,SUBSET_1:def 8; hence thesis; end; hence thesis by A2,A5,FINSEQ_3:29; end; hence thesis by A1,A4,A0,Th14; end; :: Modify POLYNOM4:17 theorem Th17: for x be Element of B holds Ext_eval(0_.A,x) = 0.B proof let x be Element of B; consider F be FinSequence of B such that A1: Ext_eval(0_.A,x) = Sum F and A2: len F = len 0_.A and for n be Element of NAT st n in dom F holds F.n = In((0_.A).(n-'1),B) * (power B).(x,n-'1) by Def1; F = <*>the carrier of B by A2,POLYNOM4:3; hence thesis by A1,RLVECT_1:43; end; :: Modify POLYNOM4:18 theorem Th18: for A,B being non degenerated Ring for x be Element of B st A is Subring of B holds Ext_eval(1_.A,x) = 1.B proof let A,B be non degenerated Ring; let x be Element of B; assume A0: A is Subring of B; consider F be FinSequence of B such that A1: Ext_eval(1_.A,x) = Sum F and A2: len F = len 1_.A and A3: for n be Element of NAT st n in dom F holds F.n = In((1_.A).(n-'1),B)*(power B).(x,n-'1) by Def1; len F = 1 by A2,POLYNOM4:4; then A4: F.1 = In((1_.A).(1-'1),B) * (power B).(x,1-'1) by A3,FINSEQ_3:25 .= In((1_.A).(0),B) * (power B).(x,1-'1) by XREAL_1:232 .= In(1.A,B) * (power B).(x,1-'1) by POLYNOM3:30 .= 1.B * (power B).(x,1-'1) by A0,Lm5 .= (power B).(x,0) by XREAL_1:232 .= 1_B by GROUP_1:def 7 .= 1.B; Sum F = Sum <*1.B*> by A2,POLYNOM4:4,FINSEQ_1:40,A4 .= 1.B by RLVECT_1:44; hence thesis by A1; end; :: Modify POLYNOM4:19 theorem Th19: for x be Element of B, p,q be Polynomial of A st A is Subring of B holds Ext_eval(p+q,x) = Ext_eval(p,x) + Ext_eval(q,x) proof let x be Element of B, p,q be Polynomial of A; assume A0: A is Subring of B; reconsider k = max(len p,len q) as Element of NAT; A1: k - len p >= 0 by XREAL_1:48,XXREAL_0:25; consider F1 be FinSequence of B such that A2: Ext_eval(p,x) = Sum F1 and A3: len F1 = len p and A4: for n be Element of NAT st n in dom F1 holds F1.n = In(p.(n-'1),B) * (power B).(x,n-'1) by Def1; A5: len (F1 ^ ((k-'(len F1)) |-> 0.B)) = len p + len ((k-'(len p)) |-> 0.B) by A3,FINSEQ_1:22 .= len p + (k-'(len p)) by CARD_1:def 7 .= len p + (k-(len p)) by A1,XREAL_0:def 2 .= k; A6: k - len q >= 0 by XREAL_1:48,XXREAL_0:25; k >= len p & k >= len q by XXREAL_0:25; then A7: k - len (p+q) >= 0 by POLYNOM4:6,XREAL_1:48; consider F3 be FinSequence of B such that A8: Ext_eval(p+q,x) = Sum F3 and A9: len F3 = len (p+q) and A10: for n be Element of NAT st n in dom F3 holds F3.n = In((p+q).(n-'1),B) * (power B).(x,n-'1) by Def1; A11: len (F3 ^ ((k-'(len F3)) |-> 0.B)) = len F3 + len((k-'(len F3)) |-> 0.B) by FINSEQ_1:22 .= len (p+q) + (k-'(len (p+q))) by CARD_1:def 7,A9 .= len (p+q) + (k-(len (p+q))) by A7,XREAL_0:def 2 .= k; consider F2 be FinSequence of B such that A12: Ext_eval(q,x) = Sum F2 and A13: len F2 = len q and A14: for n be Element of NAT st n in dom F2 holds F2.n = In(q.(n-'1),B) * (power B).(x,n-'1) by Def1; len (F2 ^ ((k-'(len F2)) |-> 0.B)) = len q + len ((k-'(len q)) |-> 0.B) by A13,FINSEQ_1:22 .= len q + (k-'(len q)) by CARD_1:def 7 .= len q + (k-(len q)) by A6,XREAL_0:def 2 .= k; then reconsider G1 = F1 ^ ((k-'(len F1)) |-> 0.B), G2 = F2 ^ ((k-'(len F2)) |-> 0.B), G3 = F3 ^ ((k-'(len F3)) |-> 0.B) as Element of k-tuples_on the carrier of B by A5,A11,FINSEQ_2:92; now let n be Nat; assume A15: n in Seg k; then A16: 0+1 <= n by FINSEQ_1:1; A17: n <= k by A15,FINSEQ_1:1; per cases by XXREAL_0:1; suppose A18: len p > len q; then k = len p by XXREAL_0:def 10; then len(p+q) = len p by A18,POLYNOM4:7; then A20: n in dom F3 by A9,A15,A18,XXREAL_0:def 10,FINSEQ_1:def 3; A21: len G2 = k by CARD_1:def 7; then A22: n in dom G2 by A15,FINSEQ_1:def 3; then A23: G2/.n = G2.n by PARTFUN1:def 6; len G1 = k by CARD_1:def 7; then A24: n in dom G1 by A15,FINSEQ_1:def 3; then A25: G1/.n = G1.n by PARTFUN1:def 6; A26: n in dom F1 by A3,A15,A18,XXREAL_0:def 10,FINSEQ_1:def 3; A27: G1/.n = G1.n by A24,PARTFUN1:def 6 .= F1.n by A26,FINSEQ_1:def 7 .= F1/.n by A26,PARTFUN1:def 6; A28: F1.n = In(p.(n-'1),B)*(power B).(x,n-'1) by A4,A26; now per cases; suppose n <= len q; then A29: n in dom F2 by A16,A13,FINSEQ_3:25; then A30: F2.n = In(q.(n-'1),B)*(power B).(x,n-'1) by A14; A31: G2/.n = G2.n by A22,PARTFUN1:def 6 .= F2.n by A29,FINSEQ_1:def 7 .= F2/.n by A29,PARTFUN1:def 6; thus G3.n = F3.n by A20,FINSEQ_1:def 7 .= In((p+q).(n-'1),B)*(power B).(x,n-'1) by A10,A20 .= In((p.(n-'1) + q.(n-'1)),B) * (power B).(x,n-'1) by NORMSP_1:def 2 .= ( In (p.(n-'1),B) + In( q.(n-'1),B)) * (power B).(x,n-'1) by A0,Th12 .= In(p.(n-'1),B)*(power B).(x,n-'1) + In(q.(n-'1),B)*(power B).(x,n-'1) by VECTSP_1:def 3 .= In(p.(n-'1),B)*(power B).(x,n-'1) + F2/.n by A29,A30,PARTFUN1:def 6 .= F1/.n + F2/.n by A26,A28,PARTFUN1:def 6 .= (G1 + G2).n by A15,A25,A23,A27,A31,FVSUM_1:18; end; suppose A32: n > len q; then A33: n >= len q+1 by NAT_1:13; then n-1 >= len q by XREAL_1:19; then A34: n-'1 >= len q by XREAL_0:def 2; n-len F2 <= k-len F2 by A17,XREAL_1:9; then A35: n-len F2 <= k-'len F2 by XREAL_0:def 2; A36: n-len F2 >= 1 by A13,A33,XREAL_1:19; then n-len F2 = n-'len F2 by XREAL_0:def 2; then A37: n-len F2 in Seg (k-'len F2) by A36,A35; n <= len G2 by A15,A21,FINSEQ_1:1; then A38: G2/.n = ((k-'(len F2)) |-> 0.B).(n - len F2) by A13,A23,A32,FINSEQ_1:24 .= 0.B by A37,FUNCOP_1:7; thus G3.n = F3.n by A20,FINSEQ_1:def 7 .= In((p+q).(n-'1),B) * (power B).(x,n-'1) by A10,A20 .= In(p.(n-'1) + q.(n-'1),B) * (power B).(x,n-'1) by NORMSP_1:def 2 .= In(p.(n-'1) + 0.A,B) * (power B).(x,n-'1) by A34,ALGSEQ_1:8 .= F1.n by A4,A26 .= G1/.n + 0.B by A26,A27,PARTFUN1:def 6 .= (G1 + G2).n by A15,A25,A23,A38,FVSUM_1:18; end; end; hence G3.n = (G1 + G2).n; end; suppose A39: len p < len q; then k = len q by XXREAL_0:def 10; then len(p+q) = len q by A39,POLYNOM4:7; then A41: n in dom F3 by A9,A15,A39,XXREAL_0:def 10,FINSEQ_1:def 3; A42: len G1 = k by CARD_1:def 7; then A43: n in dom G1 by A15,FINSEQ_1:def 3; then A44: G1/.n = G1.n by PARTFUN1:def 6; len G2 = k by CARD_1:def 7; then A45: n in dom G2 by A15,FINSEQ_1:def 3; then A46: G2/.n = G2.n by PARTFUN1:def 6; A47: n in dom F2 by A13,A15,A39,XXREAL_0:def 10,FINSEQ_1:def 3; A48: G2/.n = G2.n by A45,PARTFUN1:def 6 .= F2.n by A47,FINSEQ_1:def 7 .= F2/.n by A47,PARTFUN1:def 6; A49: F2.n = In(q.(n-'1),B)*(power B).(x,n-'1) by A14,A47; now per cases; suppose n <= len p; then A50: n in dom F1 by A16,A3,FINSEQ_3:25; then A51: F1.n = In(p.(n-'1),B)*(power B).(x,n-'1) by A4; A52: G1/.n = G1.n by A43,PARTFUN1:def 6 .= F1.n by A50,FINSEQ_1:def 7 .= F1/.n by A50,PARTFUN1:def 6; thus G3.n = F3.n by A41,FINSEQ_1:def 7 .= In((p+q).(n-'1),B) * (power B).(x,n-'1) by A10,A41 .= In(p.(n-'1) + q.(n-'1),B) * (power B).(x,n-'1) by NORMSP_1:def 2 .= ( In (p.(n-'1),B) + In( q.(n-'1),B)) * (power B).(x,n-'1) by A0,Th12 .= In(p.(n-'1),B)*(power B).(x,n-'1) + In(q.(n-'1),B)*(power B).(x,n-'1) by VECTSP_1:def 3 .= F1/.n + In(q.(n-'1),B)*(power B).(x,n-'1) by A50,A51,PARTFUN1:def 6 .= F1/.n + F2/.n by A47,A49,PARTFUN1:def 6 .= (G1 + G2).n by A15,A44,A46,A48,A52,FVSUM_1:18; end; suppose A53: n > len p; then A54: n >= len p+1 by NAT_1:13;then n-1 >= len p by XREAL_1:19; then A55: n-'1 >= len p by XREAL_0:def 2; n-len F1 <= k-len F1 by A17,XREAL_1:9; then A56: n-len F1 <= k-'len F1 by XREAL_0:def 2; A57: n-len F1 >= 1 by A3,A54,XREAL_1:19; then n-len F1 = n-'len F1 by XREAL_0:def 2; then A58: n-len F1 in Seg (k-'len F1) by A57,A56; n <= len G1 by A15,A42,FINSEQ_1:1; then A59: G1/.n = ((k-'(len F1)) |-> 0.B).(n-len F1) by A3,A44,A53,FINSEQ_1:24 .= 0.B by A58,FUNCOP_1:7; thus G3.n = F3.n by A41,FINSEQ_1:def 7 .= In((p+q).(n-'1),B)*(power B).(x,n-'1) by A10,A41 .= In(p.(n-'1) + q.(n-'1),B) * (power B).(x,n-'1) by NORMSP_1:def 2 .= In(0.A + q.(n-'1),B) * (power B).(x,n-'1) by A55,ALGSEQ_1:8 .= F2.n by A14,A47 .= 0.B + G2/.n by A47,A48,PARTFUN1:def 6 .= (G1 + G2).n by A15,A44,A46,A59,FVSUM_1:18; end; end; hence G3.n = (G1 + G2).n; end; suppose A60: len p = len q; len G2 = k by CARD_1:def 7; then A61: n in dom G2 by A15,FINSEQ_1:def 3; then A62: G2/.n = G2.n by PARTFUN1:def 6; len G1 = k by CARD_1:def 7; then A63: n in dom G1 by A15,FINSEQ_1:def 3; then A64: G1/.n = G1.n by PARTFUN1:def 6; A65: len G3 = k by CARD_1:def 7; A66: n in dom F2 by A13,A15,A60,FINSEQ_1:def 3; A67: n in dom F1 by A3,A15,A60,FINSEQ_1:def 3; then A68: F1.n = In(p.(n-'1),B)*(power B).(x,n-'1) by A4; A69: G1/.n = G1.n by A63,PARTFUN1:def 6 .= F1.n by A67,FINSEQ_1:def 7 .= F1/.n by A67,PARTFUN1:def 6; now per cases; suppose A70: n <= len (p+q); A71: n in dom F2 by A13,A15,A60,FINSEQ_1:def 3; then A72: F2.n = In(q.(n-'1),B)*(power B).(x,n-'1) by A14; A73: G2/.n = G2.n by A61,PARTFUN1:def 6 .= F2.n by A71,FINSEQ_1:def 7 .= F2/.n by A71,PARTFUN1:def 6; n in Seg len (p+q) by A16,A70; then A74: n in dom F3 by A9,FINSEQ_1:def 3; hence G3.n = F3.n by FINSEQ_1:def 7 .= In((p+q).(n-'1),B) * (power B).(x,n-'1) by A10,A74 .= In(p.(n-'1) + q.(n-'1),B)*(power B).(x,n-'1) by NORMSP_1:def 2 .= ( In (p.(n-'1),B) + In( q.(n-'1),B)) * (power B).(x,n-'1) by A0,Th12 .= In(p.(n-'1),B)*(power B).(x,n-'1) + In(q.(n-'1),B)*(power B).(x,n-'1) by VECTSP_1:def 3 .= In(p.(n-'1),B)*(power B).(x,n-'1) + F2/.n by A71,A72,PARTFUN1:def 6 .= F1/.n + F2/.n by A67,A68,PARTFUN1:def 6 .= (G1 + G2).n by A15,A64,A62,A69,A73,FVSUM_1:18; end; suppose A75: n > len (p+q); then A76: n >= len (p+q)+1 by NAT_1:13; then n-1 >= len (p+q)+1-1 by XREAL_1:9; then A77: n-'1 >= len (p+q) by XREAL_0:def 2; n-len F3 <= k-len F3 by A17,XREAL_1:9; then A78: n-len F3 <= k-'len F3 by XREAL_0:def 2; A79: G2.n = F2.n by A66,FINSEQ_1:def 7 .= In(q.(n-'1),B)*(power B).(x,n-'1) by A14,A66; A80: G1.n = F1.n by A67,FINSEQ_1:def 7 .= In(p.(n-'1),B)*(power B).(x,n-'1) by A4,A67; A81: n-len F3 >= 1 by A9,A76,XREAL_1:19; then n-len F3 = n-'len F3 by XREAL_0:def 2; then A82: n-len F3 in Seg (k-'len F3) by A81,A78; n <= len G3 by A15,A65,FINSEQ_1:1; hence G3.n=((k-'(len F3)) |-> 0.B).(n-len F3) by A9,A75,FINSEQ_1:24 .= 0.B * (power B).(x,n-'1) by A82,FUNCOP_1:7 .= In(0.A,B)*(power B).(x,n-'1) by A0,Lm5 .= In((p+q).(n-'1),B) * (power B).(x,n-'1) by A77,ALGSEQ_1:8 .= In(p.(n-'1) + q.(n-'1),B) * (power B).(x,n-'1) by NORMSP_1:def 2 .= ( In (p.(n-'1),B) + In( q.(n-'1),B)) * (power B).(x,n-'1) by A0,Th12 .= In(p.(n-'1),B)*(power B).(x,n-'1) + In(q.(n-'1),B)*(power B).(x,n-'1) by VECTSP_1:def 3 .= (G1 + G2).n by A15,A80,A79,FVSUM_1:18; end; end; hence G3.n = (G1 + G2).n; end; end; then A83: G3 = G1 + G2 by FINSEQ_2:119; A84: Sum G3 = Sum F3 + Sum ((k-'(len F3)) |-> 0.B) by RLVECT_1:41 .= Sum F3 + 0.B by MATRIX_3:11 .= Sum F3; A85: Sum G2 = Sum F2 + Sum ((k-'(len F2)) |-> 0.B) by RLVECT_1:41 .= Sum F2 + 0.B by MATRIX_3:11 .= Sum F2; Sum G1 = Sum F1 + Sum ((k-'(len F1)) |-> 0.B) by RLVECT_1:41 .= Sum F1 + 0.B by MATRIX_3:11 .= Sum F1; hence thesis by A2,A12,A8,A85,A84,A83,FVSUM_1:76; end; theorem Th20: for p,q be Polynomial of A st A is Subring of B & len p > 0 & len q > 0 for x be Element of B holds Ext_eval((Leading-Monomial(p))*'(Leading-Monomial(q)),x) = In(p.(len p-'1)*q.(len q-'1),B)*(power B).(x,len p+len q-'2) proof let p,q be Polynomial of A; assume that A0: A is Subring of B and A1: len p > 0 and A2: len q > 0; A3: len q >= 0+1 by A2,NAT_1:13; A5: len p >= 0+1 by A1,NAT_1:13; A6: len p-1 = len p-'1 by A1,XREAL_0:def 2; A7: len p + len q >= 0+(1+1) by A5,A3,XREAL_1:7; reconsider i1=len p + len q - 1 as Element of NAT by A1,INT_1:3; A9: i1-'1+1 = i1 by A7,XREAL_1:19,XREAL_1:235; set LMp=Leading-Monomial(p), LMq=Leading-Monomial(q); let x be Element of B; consider F be FinSequence of B such that A10: Ext_eval(LMp*'LMq,x) = Sum F and A11: len F = len (LMp*'LMq) and A12: for n be Element of NAT st n in dom F holds F.n=In((LMp*'LMq).(n-'1),B)*(power B).(x,n-'1) by Def1; consider r be FinSequence of A such that A13: len r = i1-'1+1 and A14: (LMp*'LMq).(i1-'1) = Sum r and A15: for k be Element of NAT st k in dom r holds r.k = LMp.(k-'1)*LMq.(i1-'1+1-'k) by POLYNOM3:def 9; len p+0 <= len p+(len q-1) by A2,XREAL_1:7; then len p in Seg len r by A5,A9,A13; then A16: len p in dom r by FINSEQ_1:def 3; i1-'len p = len p+(len q-1)-len p by A2,XREAL_0:def 2 .= len q-'1 by A2,XREAL_0:def 2; then A17: r.(len p) = LMp.(len p-'1) * LMq.(len q-'1) by A9,A15,A16; now let i be Element of NAT; assume that A18: i in dom r and A19: i <> len p; i >= 0+1 by A18,FINSEQ_3:25; then i-'1 = i-1 by XREAL_0:def 2; then A20: i-'1 <> len p-'1 by A6,A19; thus r/.i = r.i by A18,PARTFUN1:def 6 .= LMp.(i-'1) * LMq.(i1-'1+1-'i) by A15,A18 .= 0.A*LMq.(i1-'1+1-'i) by A20,POLYNOM4:def 1 .= 0.A; end; then A21: Sum r = r/.(len p) by A16,POLYNOM2:3 .= LMp.(len p-'1) * LMq.(len q-'1) by A16,A17,PARTFUN1:def 6 .= p.(len p-'1) * LMq.(len q-'1) by POLYNOM4:def 1 .= p.(len p-'1) * q.(len q-'1) by POLYNOM4:def 1; A22: len q-1 = len q-'1 by A2,XREAL_0:def 2; A23: now let i be Element of NAT; assume that A24: i in dom F and A25: i <> i1; consider r1 be FinSequence of A such that A26: len r1 = i-'1+1 and A27: (LMp*'LMq).(i-'1) = Sum r1 and A28: for k be Element of NAT st k in dom r1 holds r1.k=LMp.(k-'1)*LMq. (i-'1+1-'k) by POLYNOM3:def 9; A29: i-'1+1 = i by A24,FINSEQ_3:25,XREAL_1:235; A30: now let j be Element of NAT; assume A31: j in dom r1; then j >= 0+1 by FINSEQ_3:25; then A32: j-'1 = j-1 by XREAL_0:def 2; per cases; suppose j<>len p; then A33: j-'1 <> len p-'1 by A6,A32; thus r1.j = LMp.(j-'1)*LMq.(i-'1+1-'j) by A28,A31 .= 0.A*LMq.(i-'1+1-'j) by A33,POLYNOM4:def 1 .= 0.A; end; suppose A34: j=len p; j in Seg len r1 by A31,FINSEQ_1:def 3; then i >= 0+j by A26,A29,FINSEQ_1:1; then i-'len p = i-len p by A34,XREAL_1:19,XREAL_0:def 2; then A35: i-'len p <> len q-'1 by A22,A25; thus r1.j = LMp.(j-'1)*LMq.(i-'len p) by A28,A29,A31,A34 .= LMp.(j-'1)*0.A by A35,POLYNOM4:def 1 .= 0.A; end; end; thus F/.i = F.i by A24,PARTFUN1:def 6 .= In(Sum r1,B)*(power B).(x,i-'1) by A12,A24,A27 .= In(0.A,B)*(power B).(x,i-'1) by A30,POLYNOM3:1 .= 0.B*(power B).(x,i-'1) by A0,Lm5 .= 0.B; end; A36: len p+len q-2 >= 0 by A7,XREAL_1:19; len p+len q-(1+1) >= 0 by A7,XREAL_1:19; then A37: i1-'1 = len p+len q-1-1 by XREAL_0:def 2 .= len p+len q-'2 by A36,XREAL_0:def 2; per cases; suppose (LMp*'LMq).(i1-'1) <> 0.A; then len F >= i1 by A11,ALGSEQ_1:8,A9,NAT_1:13; then A38: i1 in dom F by A7,XREAL_1:19,FINSEQ_3:25; hence Ext_eval((Leading-Monomial(p))*'(Leading-Monomial(q)),x) = F/.i1 by A10,A23,POLYNOM2:3 .= F.i1 by A38,PARTFUN1:def 6 .= In(p.(len p-'1)*q.(len q-'1),B) *(power B).(x,len p+len q-'2) by A12,A14,A37,A21,A38; end; suppose A39: (LMp*'LMq).(i1-'1) = 0.A; now let j be Nat; assume j >= 0; j in NAT by ORDINAL1:def 12; then consider r1 be FinSequence of A such that A40: len r1 = j+1 and A41: (LMp*'LMq).j = Sum r1 and A42: for k be Element of NAT st k in dom r1 holds r1.k = LMp.(k-'1)* LMq.(j+1-'k) by POLYNOM3:def 9; now per cases; suppose j = i1-'1; hence (LMp*'LMq).j = 0.A by A39; end; suppose A43: j <> i1-'1; now let k be Element of NAT; assume A44: k in dom r1; per cases; suppose A45: k-'1 <> len p-'1; thus r1.k = LMp.(k-'1)*LMq.(j+1-'k) by A42,A44 .= 0.A*LMq.(j+1-'k) by A45,POLYNOM4:def 1 .= 0.A; end; suppose A46: k-'1 = len p-'1; A47: k in Seg len r1 by A44,FINSEQ_1:def 3; 0+1 <= k by A44,FINSEQ_3:25; then A48: k-'1 = k-1 by XREAL_0:def 2; 0+k <= j+1 by A40,A47,FINSEQ_1:1; then j+1-k >= 0 by XREAL_1:19; then A49: j+1-'k = j-len p+1 by A6,A46,A48,XREAL_0:def 2; A50: j-len p+1 <> i1-'1-len p+1 by A43; thus r1.k = LMp.(k-'1)*LMq.(j+1-'k) by A42,A44 .= LMp.(k-'1)*0.A by A22,A9,A49,A50,POLYNOM4:def 1 .= 0.A; end; end; hence (LMp*'LMq).j = 0.A by A41,POLYNOM3:1; end; end; hence (LMp*'LMq).j = 0.A; end; then 0 is_at_least_length_of (LMp*'LMq); then len (LMp*'LMq) = 0 by ALGSEQ_1:def 3; then A52: LMp*'LMq = 0_.A by POLYNOM4:5; 0.B = In(p.(len p-'1) * q.(len q-'1),B) by A0,Lm5,A21,A14,A39; hence thesis by Th17,A52; end; end; theorem Th21: for p be Polynomial of A for x be Element of B st A is Subring of B holds Ext_eval(Leading-Monomial(p),x) = In(p.(len p-'1),B) * (power B).(x,len p-'1) proof let p be Polynomial of A; let x be Element of B; assume A0: A is Subring of B; set LMp=Leading-Monomial(p); consider F be FinSequence of B such that A1: Ext_eval(LMp,x) = Sum F and A2: len F = len LMp and A3: for n be Element of NAT st n in dom F holds F.n = In(LMp.(n-'1),B)*(power B).(x,n-'1) by Def1; A4: len F = len p by A2,POLYNOM4:15; per cases; suppose A5: len p > 0; then A7: len p >= 0+1 by NAT_1:13; then A6: len p in dom F by A4,FINSEQ_3:25; now A8: len p-'1 = len p-1 by A5,XREAL_0:def 2; let i be Element of NAT; assume that A9: i in dom F and A10: i <> len p; i >= 0+1 by A9,FINSEQ_3:25; then i-'1 = i-1 by XREAL_0:def 2; then A11: i-'1 <> len p-'1 by A10,A8; thus F/.i = F.i by A9,PARTFUN1:def 6 .= In(LMp.(i-'1),B)*(power B).(x,i-'1) by A3,A9 .= In(0.A,B)*(power B).(x,i-'1) by A11,POLYNOM4:def 1 .= 0.B *(power B).(x,i-'1) by A0,Lm5 .= 0.B; end; then Sum F = F/.(len p) by A4,A7,FINSEQ_3:25,POLYNOM2:3 .= F.(len p) by A6,PARTFUN1:def 6 .= In(LMp.(len p-'1),B)*(power B).(x,len p-'1) by A3,A7,A4,FINSEQ_3:25; hence thesis by A1,POLYNOM4:def 1; end; suppose A12: len p = 0; then A13: p = 0_.A by POLYNOM4:5; LMp = 0_.A by A12,POLYNOM4:12; hence Ext_eval(Leading-Monomial(p),x) = 0.B*(power B).(x,len p-'1) by Th17 .= In(0.A,B) *(power B).(x,len p-'1) by A0,Lm5 .= In(p.(len p-'1),B)*(power B).(x,len p-'1) by A13,FUNCOP_1:7; end; end; ::Modify POLYNOM_4:Lm3: theorem Th22: for B be comRing for p,q be Polynomial of A for x be Element of B st A is Subring of B holds Ext_eval( (Leading-Monomial p)*'(Leading-Monomial q),x) = Ext_eval(Leading-Monomial(p),x)*Ext_eval(Leading-Monomial(q),x) proof let B be comRing; let p,q be Polynomial of A; let x be Element of B; assume A0: A is Subring of B; per cases; suppose A1: len p <> 0 & len q <> 0; then A2: len q >= 0+1 & len p >= 0+1 by NAT_1:13; A3: len q-1 = len q-'1 & len p-1 = len p-'1 by A1,XREAL_0:def 2; len p+len q >= 0+(1+1) by A2,XREAL_1:7; then A4: len p+len q-'2 = len p+len q-2 by XREAL_1:19,XREAL_0:def 2; A5: len p+len q-(1+1) = len p-1+(len q-1); thus Ext_eval((Leading-Monomial(p))*'(Leading-Monomial(q)),x) = In(p.(len p-'1)*q.(len q-'1),B) *(power B).(x,len p+len q-'2) by A0,A1,Th20 .= In(p.(len p-'1)*q.(len q-'1),B) *((power B).(x,len p-'1)*(power B).(x,len q-'1)) by A3,A4,A5,POLYNOM2:1 .= In(p.(len p-'1),B)*In(q.(len q-'1),B)* ((power B).(x,len p-'1)*(power B).(x,len q-'1)) by A0,Th13 .= In(p.(len p-'1),B)*(In(q.(len q-'1),B)* ((power B).(x,len p-'1)*(power B).(x,len q-'1))) by GROUP_1:def 3 .= In(p.(len p-'1),B)*((power B).(x,len p-'1)* (In(q.(len q-'1),B)*(power B).(x,len q-'1))) by GROUP_1:def 3 .= In(p.(len p-'1),B)*(power B).(x,len p-'1)* (In(q.(len q-'1),B)*(power B).(x,len q-'1)) by GROUP_1:def 3 .= In(p.(len p-'1),B)*(power B).(x,len p-'1) * Ext_eval(Leading-Monomial(q),x) by A0,Th21 .= Ext_eval(Leading-Monomial(p),x)*Ext_eval(Leading-Monomial(q),x) by A0,Th21; end; suppose len p = 0; then A6: Leading-Monomial(p) = 0_.A by POLYNOM4:12; hence Ext_eval((Leading-Monomial(p))*'(Leading-Monomial(q)),x) = Ext_eval(0_.A,x) by POLYNOM4:2 .= 0.B * Ext_eval(Leading-Monomial(q),x) by Th17 .= Ext_eval(Leading-Monomial(p),x)* Ext_eval(Leading-Monomial(q),x) by A6,Th17; end; suppose len q = 0; then len Leading-Monomial(q) = 0 by POLYNOM4:15; then A7: Leading-Monomial(q) = 0_.A by POLYNOM4:5; hence Ext_eval((Leading-Monomial(p))*'(Leading-Monomial(q)),x) = Ext_eval(0_.A,x) by POLYNOM3:34 .= Ext_eval(Leading-Monomial(p),x)*0.B by Th17 .= Ext_eval(Leading-Monomial(p),x)* Ext_eval(Leading-Monomial(q),x) by A7,Th17; end; end; :: Modify POLYNOM4:23 theorem Th15: for B be comRing for p,q be Polynomial of A for x be Element of B st A is Subring of B holds Ext_eval((Leading-Monomial p)*'q,x) = Ext_eval(Leading-Monomial(p),x) * Ext_eval(q,x) proof let B be comRing; let p1,q be Polynomial of A; let x be Element of B; assume A0: A is Subring of B; set p=Leading-Monomial(p1); defpred P[Nat] means for q be Polynomial of A holds len q = $1 implies Ext_eval(p*'q,x) = Ext_eval(p,x)*Ext_eval(q,x); A1: for k be Nat st for n be Nat st n < k holds P[n] holds P[k] proof let k be Nat; assume A2: for n be Nat st n < k holds for q be Polynomial of A holds len q = n implies Ext_eval(p*'q,x) = Ext_eval(p,x) * Ext_eval(q,x); let q be Polynomial of A; assume A3: len q = k; per cases; suppose A4: len q <> 0; set LMq = Leading-Monomial(q); consider r be Polynomial of A such that A5: len r < len q and A6: q = r+Leading-Monomial(q) and for n be Element of NAT st n < len q-1 holds r.n = q.n by A4,POLYNOM4:16; thus Ext_eval(p*'q,x)=Ext_eval(p*'r+p*'LMq,x) by A6,POLYNOM3:31 .= Ext_eval(p*'r,x) + Ext_eval(p*'LMq,x) by A0,Th19 .= Ext_eval(p,x)*Ext_eval(r,x)+Ext_eval(p*'LMq,x) by A2,A3,A5 .= Ext_eval(p,x)*Ext_eval(r,x) + Ext_eval(p,x)*Ext_eval(LMq,x) by A0,Th22 .= Ext_eval(p,x)*(Ext_eval(r,x) + Ext_eval(LMq,x)) by VECTSP_1:def 7 .= Ext_eval(p,x) * Ext_eval(q,x) by A0,A6,Th19; end; suppose len q = 0; then A7: q = 0_.A by POLYNOM4:5; hence Ext_eval(p*'q,x) = Ext_eval(0_.A,x) by POLYNOM3:34 .= Ext_eval(p,x) * 0.B by Th17 .= Ext_eval(p,x) * Ext_eval(q,x) by A7,Th17; end; end; A8: for n be Nat holds P[n] from NAT_1:sch 4(A1); len q = len q; hence thesis by A8; end; :: Modify POLYNOM4:24 theorem Th24: for B be comRing for p,q be Polynomial of A for x be Element of B st A is Subring of B holds Ext_eval(p*'q,x) = Ext_eval(p,x) * Ext_eval(q,x) proof let B be comRing; let p,q be Polynomial of A; let x be Element of B; assume A0: A is Subring of B; defpred P[Nat] means for p be Polynomial of A holds len p = $1 implies Ext_eval(p*'q,x) = Ext_eval(p,x)* Ext_eval(q,x); A1: for k be Nat st for n be Nat st n < k holds P[n] holds P[k] proof let k be Nat; assume A2: for n be Nat st n < k holds for p be Polynomial of A holds len p = n implies Ext_eval(p*'q,x) = Ext_eval(p,x) * Ext_eval(q,x); let p be Polynomial of A; assume A3: len p = k; per cases; suppose A4: len p <> 0; set LMp = Leading-Monomial(p); consider r be Polynomial of A such that A5: len r < len p and A6: p = r+Leading-Monomial(p) and for n be Element of NAT st n < len p-1 holds r.n = p.n by A4,POLYNOM4:16; thus Ext_eval(p*'q,x) = Ext_eval(r*'q+LMp*'q,x) by A6,POLYNOM3:32 .= Ext_eval(r*'q,x) + Ext_eval(LMp*'q,x) by A0,Th19 .= Ext_eval(r,x)*Ext_eval(q,x) + Ext_eval(LMp*'q,x) by A2,A3,A5 .= Ext_eval(r,x)*Ext_eval(q,x) + Ext_eval(LMp,x)*Ext_eval(q,x) by A0,Th15 .= (Ext_eval(r,x)+Ext_eval(LMp,x))*Ext_eval(q,x) by VECTSP_1:def 7 .= Ext_eval(p,x) * Ext_eval(q,x) by A0,A6,Th19; end; suppose len p = 0; then A7: p = 0_.A by POLYNOM4:5; hence Ext_eval(p*'q,x) = Ext_eval(0_.A,x) by POLYNOM4:2 .= 0.B * Ext_eval(q,x) by Th17 .= Ext_eval(p,x) * Ext_eval(q,x) by A7,Th17; end; end; A8: for n be Nat holds P[n] from NAT_1:sch 4(A1); len p = len p; hence thesis by A8; end; :: modified POLYNOM5:37 theorem Th25: for x be Element of B, z0 be Element of A st A is Subring of B holds Ext_eval(<%z0%>,x) = In(z0,B) proof let x be Element of B, z0 be Element of A; assume A0: A is Subring of B; consider F be FinSequence of B such that A1: Ext_eval(<%z0%>,x) = Sum F and A2: len F = len <%z0%> and A3: for n be Element of NAT st n in dom F holds F.n = In(<%z0%>.(n-'1),B)*(power B).(x,n-'1) by Def1; per cases by A2,ALGSEQ_1:def 5,NAT_1:25; suppose A4: len F = 0; A5: z0 = <%z0%>.0 by POLYNOM5:32 .= (0_.A).0 by A4,A2,POLYNOM4:5 .=0.A by FUNCOP_1:7; Ext_eval(<%z0%>,x) = Ext_eval(0_.A,x) by A4,A2,POLYNOM4:5 .= 0.B by Th17 .= In(z0,B) by A5,A0,Lm5; hence thesis; end; suppose A6: len F = 1; then A7: F.1 = In(<%z0%>.(1-'1),B)*(power B).(x,1-'1) by A3,FINSEQ_3:25 .=In( <%z0%>.0,B)*(power B).(x,1-'1) by XREAL_1:232 .= In( <%z0%>.0,B)*(power B).(x,0) by XREAL_1:232 .= In( z0,B) * (power B).(x,0) by POLYNOM5:32 .= In(z0,B) * 1_B by GROUP_1:def 7 .= In(z0,B); Ext_eval(<%z0%>,x) = Sum <*In(z0,B)*> by A6,A7,FINSEQ_1:40,A1 .= In(z0,B) by RLVECT_1:44; hence thesis; end; end; :: modified POLYNOM5:44 theorem for x be Element of B, z0,z1 be Element of A st A is Subring of B holds Ext_eval(<%z0,z1%>,x) = In(z0,B)+In(z1,B)*x proof let x be Element of B, z0,z1 be Element of A; assume A0: A is Subring of B; consider F be FinSequence of B such that A1: Ext_eval(<%z0,z1%>,x) = Sum F and A2: len F = len <%z0,z1%> and A3: for n be Element of NAT st n in dom F holds F.n = In(<%z0,z1%>.(n-'1),B)*(power B).(x,n-'1) by Def1; len F = 0 or ... or len F = 2 by A2,POLYNOM5:39; then per cases; suppose len F = 0; then A4: <%z0,z1%> = 0_.A by A2,POLYNOM4:5; hence Ext_eval(<%z0,z1%>,x)=0.B by Th17.=In(0.A,B) by A0,Lm5 .= In((0_.A).0,B) by FUNCOP_1:7 .= In(z0,B)+ 0.B * x by A4,POLYNOM5:38 .= In(z0,B) + In(0.A,B) *x by A0,Lm5 .= In(z0,B) + In((0_.A).1,B)*x by FUNCOP_1:7 .= In(z0,B)+ In(z1,B)*x by A4,POLYNOM5:38; end; suppose A5: len F = 1; then F.1 = In(<%z0,z1%>.(1-'1),B)*(power B).(x,1-'1) by A3,FINSEQ_3:25 .= In(<%z0,z1%>.0,B)* (power B).(x,1-'1) by XREAL_1:232 .= In(<%z0,z1%>.0,B)* (power B).(x,0) by XREAL_1:232 .= In(z0,B) * (power B).(x,0) by POLYNOM5:38 .= In(z0,B) * 1_B by GROUP_1:def 7 .= In(z0,B); then F = <*In(z0,B)*> by A5,FINSEQ_1:40; hence Ext_eval(<%z0,z1%>,x) = In(z0,B) + 0.B*x by A1,RLVECT_1:44 .= In(z0,B) + In(0.A,B) *x by A0,Lm5 .= In(z0,B) + In(<%z0,z1%>.1,B)*x by A2,A5,ALGSEQ_1:8 .= In(z0,B) + In(z1,B)*x by POLYNOM5:38; end; suppose A6: len F = 2; then A7: F.1 = In(<%z0,z1%>.(1-'1),B)*(power B).(x,1-'1) by A3,FINSEQ_3:25 .= In(<%z0,z1%>.0,B)* (power B).(x,1-'1) by XREAL_1:232 .= In(<%z0,z1%>.0,B)* (power B).(x,0) by XREAL_1:232 .= In(z0,B) * (power B).(x,0) by POLYNOM5:38 .= In(z0,B) * 1_B by GROUP_1:def 7 .= In(z0,B); A8: 2-'1 = 2-1 by XREAL_0:def 2; F.2 = In(<%z0,z1%>.(2-'1),B)*(power B).(x,2-'1) by A3,A6,FINSEQ_3:25 .= In(z1,B) * (power B).(x,1) by A8,POLYNOM5:38 .= In(z1,B) * x by GROUP_1:50; then F = <* In(z0,B),In(z1,B)*x *> by A6,A7,FINSEQ_1:44; hence thesis by A1,RLVECT_1:45; end; end; begin :: Definition of Integral Element over A in B definition let A,B be Ring; let x be Element of B; pred x is_integral_over A means ex f be Polynomial of A st LC f = 1.A & Ext_eval(f,x) = 0.B; end; theorem Th27: for A being non degenerated Ring for a be Element of A st A is Subring of B holds In(a,B) is_integral_over A proof let A be non degenerated Ring; let a be Element of A; assume A0: A is Subring of B; set p = <% -a, 1.A %>; p.(len p -' 1) = p.(2-'1) by POLYNOM5:40 .= p.(2-1) by XREAL_1:233 .= p.1; then A2: LC p = 1.A by POLYNOM5:38; A3: eval(p,a) = -a + a by POLYNOM5:47 .= a - a .= 0.A by RLVECT_1:15; Ext_eval(p,In(a,B)) = In(eval(p,a),B) by A0,Th16 .= 0.B by A0,Lm5,A3; hence thesis by A2; end; definition let A be non degenerated Ring, B be Ring; assume A0: A is Subring of B; func integral_closure(A,B) -> non empty Subset of B equals {z where z is Element of B: z is_integral_over A}; coherence proof set M ={z where z is Element of B: z is_integral_over A}; A1: now let u be object; assume u in M; then ex z being Element of B st u = z & z is_integral_over A; hence u in the carrier of B; end; In(0.A,B) is_integral_over A by A0,Th27; then 0.B is_integral_over A by A0,Lm5; then 0.B in M; hence thesis by A1,TARSKI:def 3; end; end; definition let c be Complex; attr c is algebraic means ex x being Element of F_Complex st x = c & x is_integral_over F_Rat; end; definition let x be Element of F_Complex; redefine attr x is algebraic means x is_integral_over F_Rat; compatibility; end; definition let c be Complex; attr c is algebraic_integer means ex x being Element of F_Complex st x = c & x is_integral_over INT.Ring; end; definition let x be Element of F_Complex; redefine attr x is algebraic_integer means x is_integral_over INT.Ring; compatibility; end; notation let x be Complex; antonym x is transcendental for x is algebraic; end; registration cluster rational -> algebraic for Complex; coherence proof let c be Complex; assume c is rational; then reconsider c as Element of F_Rat by RAT_1:def 2; take In(c,F_Complex); thus thesis by Th3,Th27; end; end; registration cluster algebraic for Complex; existence by GAUSSINT:14; cluster algebraic for Element of F_Complex; existence by Lm7; end; registration cluster integer -> algebraic_integer for Complex; coherence proof let c be Complex; assume c is integer; then reconsider c as Element of INT.Ring by INT_1:def 2; take In(c,F_Complex); thus thesis by Th4,Th27; end; end; registration cluster algebraic_integer for Complex; existence by GAUSSINT:14; cluster algebraic_integer for Element of F_Complex; existence by Lm7; end; definition let A,B be Ring; let x be Element of B; func Ann_Poly(x,A) -> non empty Subset of Polynom-Ring A equals {p where p is Polynomial of A: Ext_eval(p,x) = 0.B}; coherence proof set M ={p where p is Polynomial of A:Ext_eval(p,x) = 0.B}; A1: now let u be object; assume u in M; then ex p1 being Polynomial of A st u = p1 & Ext_eval(p1,x)=0.B; hence u in the carrier of Polynom-Ring A by POLYNOM3:def 10; end; Ext_eval(0_.A,x) = 0.B by Th17; then 0_.A in M; hence thesis by A1,TARSKI:def 3; end; end; theorem Lm30: for A,B be Ring, w be Element of B, x, y being Element of Polynom-Ring A st A is Subring of B & x in Ann_Poly(w,A) & y in Ann_Poly(w,A) holds x + y in Ann_Poly(w,A) proof let A,B; let w be Element of B; let x,y be Element of Polynom-Ring A; assume that A0: A is Subring of B and A1: x in Ann_Poly(w,A) and A2: y in Ann_Poly(w,A); reconsider x1=x, y1=y as Polynomial of A by POLYNOM3:def 10; set M ={p where p is Polynomial of A:Ext_eval(p,w)=0.B}; consider x2 be Polynomial of A such that A3: x2 = x1 and A4: Ext_eval(x2,w)=0.B by A1; consider y2 be Polynomial of A such that A5: y2 = y1 and A6: Ext_eval(y2,w)=0.B by A2; A7: Ext_eval(x2 + y2,w) = Ext_eval(x1,w) + 0.B by A0,Th19,A6,A3 .= 0.B by A3,A4; consider t be Polynomial of A such that A8: t = x1+y1 and A9: Ext_eval(t,w) = 0.B by A3,A5,A7; x1+ y1 in M by A8,A9; hence thesis by POLYNOM3:def 10; end; theorem Th31: for B be comRing, z be Element of B, p, x being Element of Polynom-Ring A st A is Subring of B & x in Ann_Poly(z,A) holds p * x in Ann_Poly(z,A) proof let B be comRing; let w be Element of B; let p,x be Element of Polynom-Ring A; assume that A0: A is Subring of B and A1: x in Ann_Poly(w,A); set M ={p where p is Polynomial of A:Ext_eval(p,w)=0.B}; reconsider p1=p, x1=x as Polynomial of A by POLYNOM3:def 10; consider x2 be Polynomial of A such that A2: x2 = x1 and A3: Ext_eval(x2,w)=0.B by A1; Ext_eval(p1 *'x1,w) = Ext_eval(p1,w) * 0.B by A0,A2,A3,Th24.= 0.B; then consider t be Polynomial of A such that A4: t = p1 *'x1 and A5: Ext_eval(t,w) = 0.B; p1 *'x1 in M by A4,A5; hence thesis by POLYNOM3:def 10; end; theorem Lm32: for B be comRing for w be Element of B, p, x being Element of Polynom-Ring A st A is Subring of B & x in Ann_Poly(w,A) holds x * p in Ann_Poly(w,A) proof let B be comRing; let w be Element of B; let p,x be Element of Polynom-Ring A; set M ={p where p is Polynomial of A:Ext_eval(p,w)=0.B}; reconsider p1=p, x1=x as Polynomial of A by POLYNOM3:def 10; assume that A0: A is Subring of B and A1: x in Ann_Poly(w,A); consider x2 be Polynomial of A such that A2: x2 = x1 and A3: Ext_eval(x2,w)=0.B by A1; Ext_eval(x1*'p1,w) = Ext_eval(p1,w) * 0.B by A0,A2,A3,Th24 .= 0.B; then consider t be Polynomial of A such that A4: t = x1 *'p1 and A5: Ext_eval(t,w) = 0.B; x1 *'p1 in M by A4,A5; hence thesis by POLYNOM3:def 10; end; theorem Th33: for A be non degenerated Ring for B be non degenerated comRing for w be Element of B st A is Subring of B holds Ann_Poly(w,A) is proper Ideal of Polynom-Ring A proof let A be non degenerated Ring; let B be non degenerated comRing; let w be Element of B; assume A0: A is Subring of B; A1: Ann_Poly(w,A) is add-closed by A0,Lm30; A2: Ann_Poly(w,A) is left-ideal by A0,Th31; A3: Ann_Poly(w,A) is right-ideal by A0,Lm32; Ann_Poly(w,A) is proper proof assume not Ann_Poly(w,A) is proper; then A5: 1.Polynom-RingA in Ann_Poly(w,A); A6: 1_.A in Ann_Poly(w,A) by A5,POLYNOM3:37; A7: Ext_eval(1_.A,w)= 1.B by A0,Th18; ex p be Polynomial of A st p = 1_.A & Ext_eval(p,w)= 0.B by A6; hence contradiction by A7; end; hence thesis by A1,A2,A3; end; begin :: Properties of Polynomial Ring over PID. reserve K, L for Field; theorem Th34: for K,L be Field, w be Element of L st K is Subring of L holds ex g be Element of Polynom-Ring K st {g}-Ideal = Ann_Poly(w,K) proof let K,L; let w be Element of L; assume A0: K is Subring of L; A1: Polynom-Ring K is PID; Ann_Poly(w,K) is Ideal of Polynom-Ring K by A0,Th33; hence thesis by A1,IDEAL_1:def 27; end; theorem Th35: for K,L be Field, z be Element of L st z is_integral_over K holds Ann_Poly(z,K) <> {0.Polynom-Ring K} proof let K,L; let z be Element of L; assume A1: z is_integral_over K; set M = {p where p is Polynomial of K:Ext_eval(p,z)=0.L}; consider f be Polynomial of K such that A2: LC f = 1.K and A3: Ext_eval(f,z) = 0.L by A1; not f in {0.Polynom-Ring K} proof assume A5: f in {0.Polynom-Ring K}; reconsider f as Element of Polynom-Ring K by POLYNOM3:def 10; f in {0.Polynom-Ring K}-Ideal by A5,IDEAL_1:47; then f in the set of all 0.Polynom-Ring K*g where g is Element of Polynom-Ring K by IDEAL_1:64; then consider g1 being Element of Polynom-Ring K such that A6: f = 0.Polynom-Ring K * g1; reconsider g2 = g1 as Polynomial of K by POLYNOM3:def 10; reconsider h2 = 0.Polynom-Ring K as Polynomial of K by POLYNOM3:def 10; f = 0_.K by POLYNOM3:def 10,A6; hence contradiction by FUNCOP_1:7,A2; end; hence thesis by A3; end; theorem Lm37: for K be Field, p be Element of Polynom-Ring K st p <> 0_.K holds p is non zero Element of the carrier of Polynom-Ring K proof let K; let p be Element of Polynom-Ring K; assume A0: p <> 0_.K; assume A1: not(p is non zero Element of the carrier of Polynom-Ring K); reconsider p as Element of the carrier of Polynom-Ring K; p is zero by A1; hence contradiction by A0; end; theorem Th38: for K,L be Field, w be Element of L st K is Subring of L holds Ann_Poly(w,K) is quasi-prime proof let K,L; let w be Element of L; assume A0: K is Subring of L; set M = {p where p is Polynomial of K:Ext_eval(p,w)=0.L}; for p, q being Element of Polynom-Ring K st p*q in Ann_Poly(w,K) holds p in Ann_Poly(w,K) or q in Ann_Poly(w,K) proof let p, q be Element of Polynom-Ring K; assume A1: p*q in Ann_Poly(w,K); reconsider p1=p, q1=q as Polynomial of K by POLYNOM3:def 10; p1*'q1 in Ann_Poly(w,K) by A1,POLYNOM3:def 10; then consider t be Polynomial of K such that A5: t = p1*'q1 and A6: Ext_eval(t,w)=0.L; Ext_eval(p1,w) * Ext_eval(q1,w) = 0.L by A0,Th24,A6,A5; then per cases by VECTSP_2:def 1; suppose Ext_eval(p1,w)=0.L; hence thesis; end; suppose Ext_eval(q1,w)=0.L; hence thesis; end; end; hence thesis by RING_1:def 1; end; theorem Th39: for K,L be Field, w be Element of L st K is Subring of L & w is_integral_over K holds Ann_Poly(w,K) is prime proof let K,L; let w be Element of L; assume K is Subring of L & w is_integral_over K; then Ann_Poly(w,K) is proper quasi-prime by Th38,Th33; hence thesis; end; theorem Th40: for K,L be Field, z be Element of L st K is Subring of L & z is_integral_over K ex f be Element of Polynom-Ring K st f <> 0_.K & {f}-Ideal = Ann_Poly(z,K) & f = NormPolynomial(f) proof let K,L be Field; let z be Element of L; assume that A0: K is Subring of L and A1: z is_integral_over K; consider f be Element of Polynom-Ring K such that A2: {f}-Ideal = Ann_Poly(z,K) by A0,Th34; A3: f <> 0.Polynom-Ring K by A1,A2,Th35,IDEAL_1:47; reconsider f as Element of Polynom-Ring K; A4: f <> 0_.K by A3,POLYNOM3:def 10; set g = NormPolynomial(f); A7: {g}-Ideal = Ann_Poly(z,K) by A2,RING_4:27,RING_2:21; g <> 0.Polynom-Ring K by A1,A7,Th35,IDEAL_1:47; then A8: g <> 0_.K by POLYNOM3:def 10; then A9: g is non zero Element of the carrier of Polynom-Ring K by Lm37; A10:f is non zero Element of the carrier of Polynom-Ring K by A4,Lm37; g = NormPolynomial(g) by A9,A10,RING_4:24; hence thesis by A7,A8; end; theorem Th41: for K,L be Field, z be Element of L,f,g be Element of Polynom-Ring K st z is_integral_over K & {f}-Ideal = Ann_Poly(z,K) & f = NormPolynomial(f) & {g}-Ideal = Ann_Poly(z,K) & g = NormPolynomial(g) holds f = g proof let K,L be Field; let z be Element of L; let f,g be Element of Polynom-Ring K; assume that A1: z is_integral_over K and A2: {f}-Ideal = Ann_Poly(z,K) and A3: f = NormPolynomial(f) and A4: {g}-Ideal = Ann_Poly(z,K) and A5: g = NormPolynomial(g); reconsider f as Element of the carrier of Polynom-Ring K; NormPolynomial(f) <> 0.(Polynom-Ring K) by A3,A2,A1,Th35,IDEAL_1:47; then f <> 0_.K by A3,POLYNOM3:def 10; then A6: f is non zero Element of the carrier of Polynom-Ring K by Lm37; reconsider g as Element of the carrier of Polynom-Ring K; NormPolynomial(g) <> 0.(Polynom-Ring K) by A5,A4,A1,Th35,IDEAL_1:47; then g <> 0_.K by A5,POLYNOM3:def 10; then g is non zero Element of the carrier of Polynom-Ring K by Lm37; hence thesis by A3,A6,A5,RING_2:21,RING_4:30,A4,A2; end; definition let K,L be Field; let z be Element of L; assume that A1: K is Subring of L and A2: z is_integral_over K; func minimal_polynom(z,K) -> Element of the carrier of Polynom-Ring K means :Def7: it <> 0_.K & {it}-Ideal = Ann_Poly(z,K) & it = NormPolynomial(it); existence by A1,A2,Th40; uniqueness by Th41,A2; end; definition let K,L be Field; let z be Element of L; assume that A1: K is Subring of L and A2: z is_integral_over K; func deg_of_integral_element(z,K) -> Element of NAT equals deg (minimal_polynom(z,K)); coherence proof set f = minimal_polynom(z,K); A7: f is non zero by A1,A2,Def7; reconsider f as Polynomial of K; deg f <> -1 by A7; then A8: len f <> 0; len f + 1 > 0 + 1 by A8,XREAL_1:8; then len f >= 1 by NAT_1:13; hence thesis by INT_1:3; end; end; definition let A,B be Ring; let x be Element of B; func hom_Ext_eval(x,A) -> Function of Polynom-Ring A,B means :Def9: for p be Polynomial of A holds it.p = Ext_eval(p,x); existence proof defpred P[set,set] means ex p be Polynomial of A st p = $1 & $2 = Ext_eval(p,x); A1: for y be Element of the carrier of Polynom-Ring A ex z be Element of B st P[y,z] proof let y be Element of the carrier of Polynom-Ring A; reconsider p=y as Polynomial of A; take Ext_eval(p,x); take p; thus thesis; end; consider f be Function of Polynom-Ring A, B such that A2: for y be Element of Polynom-Ring A holds P[y,f.y] from FUNCT_2:sch 3 (A1); reconsider f as Function of Polynom-Ring A, B; take f; let p be Polynomial of A; p in Polynom-Ring A by POLYNOM3:def 10; then ex q be Polynomial of A st q = p & f.p = Ext_eval(q,x) by A2; hence thesis; end; uniqueness proof let f1,f2 be Function of Polynom-Ring A, B such that A3: for p be Polynomial of A holds f1.p = Ext_eval(p,x) and A4: for p be Polynomial of A holds f2.p = Ext_eval(p,x); now let y be Element of Polynom-Ring A; reconsider p=y as Polynomial of A by POLYNOM3:def 10; thus f1.y = Ext_eval(p,x) by A3 .= f2.y by A4; end; hence f1 = f2; end; end; registration let x be Element of F_Complex; cluster hom_Ext_eval(x,F_Rat) -> unity-preserving additive multiplicative; coherence proof thus (hom_Ext_eval(x,F_Rat)).(1_Polynom-Ring F_Rat) = (hom_Ext_eval(x,F_Rat)).(1_.(F_Rat)) by POLYNOM3:37 .= Ext_eval(1_.(F_Rat),x) by Def9 .= 1_F_Complex by Th3,Th18; hereby let a,b be Element of Polynom-Ring F_Rat; reconsider p=a,q=b as Polynomial of F_Rat by POLYNOM3:def 10; thus hom_Ext_eval(x,F_Rat).(a+b) = hom_Ext_eval(x,F_Rat).(p+q) by POLYNOM3:def 10 .= Ext_eval(p+q,x) by Def9 .= Ext_eval(p,x) + Ext_eval(q,x) by Th3,Th19 .= hom_Ext_eval(x,F_Rat).a + Ext_eval(q,x) by Def9 .= hom_Ext_eval(x,F_Rat).a + hom_Ext_eval(x,F_Rat).b by Def9; end; hereby let a,b be Element of Polynom-Ring F_Rat; reconsider p=a,q=b as Polynomial of F_Rat by POLYNOM3:def 10; thus (hom_Ext_eval(x,F_Rat)).(a*b) = (hom_Ext_eval(x,F_Rat)).(p*'q) by POLYNOM3:def 10 .= Ext_eval(p*'q,x) by Def9 .= Ext_eval(p,x)* Ext_eval(q,x) by Th3,Th24 .= (hom_Ext_eval(x,F_Rat)).a*Ext_eval(q,x) by Def9 .= (hom_Ext_eval(x,F_Rat)).a*(hom_Ext_eval(x,F_Rat)).b by Def9; end; end; end; theorem for x be Element of F_Complex holds F_Complex is (Polynom-Ring F_Rat)-homomorphic proof let x be Element of F_Complex; hom_Ext_eval(x,F_Rat) is RingHomomorphism; hence thesis; end; theorem Lm45: for x be Element of B, z be object st z in rng hom_Ext_eval(x,A) holds z in B; definition let x be Element of F_Complex; func FQ(x) -> Subset of F_Complex equals rng hom_Ext_eval(x,F_Rat); coherence; end; registration let x be Element of F_Complex; cluster FQ(x) -> non empty; coherence; end; theorem Lm46: for x,z1,z2 be Element of F_Complex st z1 in FQ(x) & z2 in FQ(x) holds z1 + z2 in FQ(x) proof let x,z1,z2 be Element of F_Complex; assume that A1: z1 in FQ(x) and A2: z2 in FQ(x); consider f1 be object such that A3: f1 in dom hom_Ext_eval(x,F_Rat) and A4: z1 = hom_Ext_eval(x,F_Rat).f1 by A1,FUNCT_1:def 3; consider f2 be object such that A5: f2 in dom hom_Ext_eval(x,F_Rat) and A6: z2 = hom_Ext_eval(x,F_Rat).f2 by A2,FUNCT_1:def 3; A7: dom hom_Ext_eval(x,F_Rat) = the carrier of Polynom-Ring F_Rat by FUNCT_2:def 1; reconsider g1 = f1, g2 = f2 as Polynomial of F_Rat by A3,A5,POLYNOM3:def 10; A8: z1 + z2 = Ext_eval(g1,x) + hom_Ext_eval(x,F_Rat).f2 by Def9,A6,A4 .= Ext_eval(g1,x) + Ext_eval(g2,x) by Def9 .= Ext_eval(g1+g2,x) by Th3,Th19 .= hom_Ext_eval(x,F_Rat).(g1+g2) by Def9; set g = g1+g2; g in dom hom_Ext_eval(x,F_Rat) by A7,POLYNOM3:def 10; hence thesis by A8,FUNCT_1:def 3; end; theorem Lm47: for x,z1,z2 be Element of F_Complex st z1 in FQ(x) & z2 in FQ(x) holds z1 * z2 in FQ(x) proof let x,z1,z2 be Element of F_Complex; assume that A1: z1 in FQ(x) and A2: z2 in FQ(x); consider f1 be object such that A3: f1 in dom hom_Ext_eval(x,F_Rat) and A4: z1 = hom_Ext_eval(x,F_Rat).f1 by A1,FUNCT_1:def 3; consider f2 be object such that A5: f2 in dom hom_Ext_eval(x,F_Rat) and A6: z2 = hom_Ext_eval(x,F_Rat).f2 by A2,FUNCT_1:def 3; A7: dom hom_Ext_eval(x,F_Rat) = the carrier of Polynom-Ring F_Rat by FUNCT_2:def 1; reconsider g1 = f1, g2 = f2 as Polynomial of F_Rat by A3,A5,POLYNOM3:def 10; A8: z1 * z2 = Ext_eval(g1,x) * hom_Ext_eval(x,F_Rat).f2 by Def9,A6,A4 .= Ext_eval(g1,x) * Ext_eval(g2,x) by Def9 .= Ext_eval(g1*'g2,x) by Th3,Th24 .= hom_Ext_eval(x,F_Rat).(g1*'g2) by Def9; set g = g1*'g2; g in dom hom_Ext_eval(x,F_Rat) by A7,POLYNOM3:def 10; hence thesis by A8,FUNCT_1:def 3; end; theorem Lm48: for x be Element of F_Complex, a be Element of F_Rat holds a in FQ(x) proof let x be Element of F_Complex; let a be Element of F_Rat; reconsider f = <% a %> as Polynomial of F_Rat; A2: dom hom_Ext_eval(x,F_Rat) = the carrier of Polynom-Ring F_Rat by FUNCT_2:def 1; A3: Ext_eval(f,x) = In(a,F_Complex) by Th3,Th25; reconsider f as Element of Polynom-Ring F_Rat by POLYNOM3:def 10; In(a,F_Complex) = hom_Ext_eval(x,F_Rat).f by A3,Def9; hence thesis by A2,FUNCT_1:def 3; end; definition let x be Element of F_Complex; func FQ_add(x) -> BinOp of FQ(x) equals addcomplex || FQ(x); correctness proof set ad = addcomplex||FQ(x); set theCFComplex = the carrier of F_Complex; A0: [:FQ(x),FQ(x):] c= [:theCFComplex,theCFComplex:]; theCFComplex = COMPLEX by COMPLFLD:def 1; then [:FQ(x),FQ(x):] c= dom (addcomplex) by A0,FUNCT_2:def 1; then A1: dom ad = [:FQ(x),FQ(x):] by RELAT_1:62; for z be object st z in [:FQ(x),FQ(x):] holds ad. z in FQ(x) proof let z be object such that A2: z in [:FQ(x),FQ(x):]; consider a, b be object such that A3: a in FQ(x) and A4: b in FQ(x) and A5: z = [a,b] by A2,ZFMISC_1:def 2; reconsider x1 = a, y1 = b as Element of theCFComplex by A3,A4; ad.z = addcomplex.(a,b) by A2,A5,FUNCT_1:49 .= x1+y1 by BINOP_2:def 3; hence ad.z in FQ(x) by A3,A4,Lm46; end; hence thesis by A1,FUNCT_2:3; end; end; definition let x be Element of F_Complex; func FQ_mult(x) -> BinOp of FQ(x) equals multcomplex || FQ(x); correctness proof set mult = multcomplex||FQ(x); set theCFComplex = the carrier of F_Complex; A0: theCFComplex = COMPLEX by COMPLFLD:def 1; [:FQ(x),FQ(x):] c= [:COMPLEX,COMPLEX:] by A0; then [:FQ(x),FQ(x):] c= dom(multcomplex) by FUNCT_2:def 1; then A1: dom mult = [:FQ(x),FQ(x):] by RELAT_1:62; for z be object st z in [:FQ(x),FQ(x):] holds mult.z in FQ(x) proof let z be object such that A2: z in [:FQ(x),FQ(x):]; consider x1,y1 be object such that A3: x1 in FQ(x) & y1 in FQ(x) & z = [x1,y1] by A2,ZFMISC_1:def 2; reconsider x2 = x1, y2 = y1 as Element of theCFComplex by A3; mult.z = multcomplex.(x2,y2) by A2,A3,FUNCT_1:49 .= x2*y2 by BINOP_2:def 5; hence mult.z in FQ(x) by A3,Lm47; end; hence thesis by A1,FUNCT_2:3; end; end; theorem Th49: for x be Element of F_Complex, z, w be Element of FQ(x) holds (FQ_add(x)).(z,w) = z+w proof let x be Element of F_Complex; let z, w be Element of FQ(x); thus (FQ_add(x)).(z,w) = addcomplex.(z,w) by FUNCT_1:49,ZFMISC_1:87 .= z+w by BINOP_2:def 3; end; theorem Th50: for x be Element of F_Complex, z, w be Element of FQ(x) holds (FQ_mult(x)).(z,w) = z*w proof let x be Element of F_Complex; let z, w be Element of FQ(x); thus (FQ_mult(x)).(z,w) = multcomplex.(z,w) by FUNCT_1:49,ZFMISC_1:87 .= z*w by BINOP_2:def 5; end; theorem Lm52: :::????? for x be Element of F_Complex holds In(1.F_Complex, FQ(x)) = 1.F_Complex proof let x be Element of F_Complex; 1.F_Complex = 1.F_Rat by C0SP1:def 3,Th3; hence thesis by Lm48,SUBSET_1:def 8; end; theorem Lm53: In(-1.F_Rat,F_Complex) = -1.F_Complex proof 1.F_Complex + In(-1.F_Rat,F_Complex) = In(1.F_Rat,F_Complex) + In(-1.F_Rat,F_Complex) by Lm5,Th3 .= In(0.F_Rat,F_Complex) .= 0.F_Complex by Lm5,Th3; hence thesis by RLVECT_1:def 10; end; definition let x be Element of F_Complex; func FQ_Ring(x) -> strict non empty doubleLoopStr equals doubleLoopStr(# FQ(x), FQ_add(x), FQ_mult(x),In(1.F_Complex,FQ(x)), In(0.F_Complex,FQ(x)) #); coherence; end; theorem Th54: for x be Element of F_Complex holds FQ_Ring(x) is Ring proof let x be Element of F_Complex; reconsider ZS = doubleLoopStr(# FQ(x),FQ_add(x),FQ_mult(x), In(1.F_Complex,FQ(x)),In(0.F_Complex,FQ(x)) #) as non empty doubleLoopStr; A1:for v, w being Element of ZS holds v + w = w + v proof let v, w be Element of ZS; v in F_Complex & w in F_Complex by Lm45; then reconsider v1 = v, w1 = w as Element of F_Complex; thus v + w = w1 + v1 by Th49 .= w + v by Th49; end; A2: for u, v, w being Element of ZS holds (u + v) + w = u + (v + w) proof let u, v, w be Element of ZS; u in F_Complex & v in F_Complex & w in F_Complex by Lm45; then reconsider u1 = u, v1 = v, w1 = w as Element of F_Complex; A3: u + v = u1+v1 by Th49; A4: v + w = v1+w1 by Th49; thus (u + v) + w = u1+v1+w1 by Th49,A3 .= u1+(v1+w1) .= u+(v+w) by Th49,A4; end; A5: for v being Element of ZS holds v + 0.ZS = v proof let v be Element of ZS; A6: 0.ZS = 0.F_Complex by Lm48,Lm7,SUBSET_1:def 8; 0.ZS in F_Complex & v in F_Complex by Lm45; then reconsider v1 = v as Element of F_Complex; thus v + 0.ZS = v1 + 0.F_Complex by Th49,A6 .= v; end; A7: for v being Element of ZS holds v is right_complementable proof let v be Element of ZS; v in F_Complex by Lm45; then reconsider v1 = v as Element of F_Complex; A8: (-1.F_Complex) * v1 = -(1.F_Complex * v1) by VECTSP_1:9 .= -v1; reconsider w1 = -1.F_Complex as Element of ZS by Lm48,Lm53; A10: w1 * v = (-1.F_Complex ) * v1 by Th50; take w1*v; thus v + (w1*v) = v1 + ((-1.F_Complex ) * v1) by A10,Th49 .= 0.F_Complex by RLVECT_1:5,A8 .= 0.ZS by Lm48,Lm7,SUBSET_1:def 8; end; A11: for a, b, v being Element of ZS holds (a + b) * v = a * v + b * v proof let a, b, v be Element of ZS; a in F_Complex & b in F_Complex & v in F_Complex by Lm45; then reconsider a1 = a, b1 = b, v1 = v as Element of F_Complex; A12: a+b in FQ(x); reconsider ab = a+b as Element of F_Complex by A12; A13: a1*v1 = a*v & (b1*v1 = b*v) by Th50; thus (a + b) * v = ab * v1 by Th50 .= (a1 + b1) * v1 by Th49 .= a1*v1 + b1*v1 .= a*v + b*v by A13,Th49; end; A14: for a, v, w being Element of ZS holds a * (v + w) = a * v + a * w & (v + w)*a = v*a + w*a proof let a, v, w be Element of ZS; a in F_Complex & v in F_Complex & w in F_Complex by Lm45; then reconsider a1 = a, v1 = v, w1 = w as Element of F_Complex; A15: v+w in FQ(x); reconsider vw = (v+w) as Element of F_Complex by A15; A16: (a1*v1 = a*v) & (a1*w1 = a*w) by Th50; thus a * (v + w) = a1 * vw by Th50 .= a1 * (v1 + w1) by Th49 .= a1*v1 + a1*w1 .= a*v + a*w by A16,Th49; thus (v + w) * a = v*a + w*a by A11; end; A17: for a, b, v being Element of ZS holds (a * b) * v = a * (b * v) proof let a, b, v be Element of ZS; a in F_Complex & b in F_Complex & v in F_Complex by Lm45; then reconsider a1 = a, b1 = b, v1 = v as Element of F_Complex; A18: a*b in FQ(x) & b*v in FQ(x); reconsider ab = (a*b), bv = (b*v) as Element of F_Complex by A18; thus (a * b) * v = ab * v1 by Th50 .= (a1 * b1) * v1 by Th50 .= a1 * (b1 * v1) .= a1 * bv by Th50 .= a * (b * v) by Th50; end; for v being Element of ZS holds v *1.ZS = v & 1.ZS * v = v proof let v be Element of ZS; A19: 1.ZS = 1.F_Complex by Lm52; v in F_Complex by Lm45; then reconsider v1 = v as Element of F_Complex; thus v * 1.ZS = v1 * 1.F_Complex by A19,Th50 .= v; thus 1.ZS * v = 1.F_Complex * v1 by A19,Th50 .= v; end; hence thesis by A1,A2,A5,A7,A14,A17,VECTSP_1:def 6,def 7, GROUP_1:def 3,RLVECT_1:def 2,def 3,def 4,ALGSTR_0:def 16; end; registration let x be Element of F_Complex; cluster FQ_Ring(x) -> Abelian add-associative right_zeroed right_complementable associative well-unital distributive; coherence by Th54; end; registration let z be Element of F_Complex; cluster FQ_Ring(z) -> domRing-like commutative non degenerated; coherence proof set X = FQ_Ring(z); thus X is domRing-like proof for x, y being Element of X holds x*y = 0.X implies x = 0.X or y = 0.X proof let x, y be Element of X; x in F_Complex & y in F_Complex by Lm45; then reconsider x1 = x, y1 = y as Element of F_Complex; assume A1: x*y = 0.X; A2: 0.X = 0.F_Complex by Lm48,Lm7,SUBSET_1:def 8; then x1*y1 = 0.F_Complex by A1,Th50; hence thesis by A2,VECTSP_1:12; end; hence thesis by VECTSP_2:def 1; end; A3: 0.FQ_Ring(z) = 0.F_Complex by Lm48,Lm7,SUBSET_1:def 8; thus X is commutative proof let v, w be Element of FQ_Ring(z); v in F_Complex & w in F_Complex by Lm45; then reconsider v1=v, w1=w as Element of F_Complex; thus v * w = v1 * w1 by Th50 .= w * v by Th50; end; thus thesis by Lm52,A3; end; end; Lm55: for x be Element of F_Complex holds the carrier of F_Rat c= the carrier of FQ_Ring(x) by Lm48; theorem Lm56: for x be Element of F_Complex holds [:RAT,RAT:] c= [:FQ(x),FQ(x):] & [:FQ(x),FQ(x):] c= [:COMPLEX,COMPLEX:] proof let x be Element of F_Complex; A1: the carrier of F_Rat c= the carrier of FQ_Ring(x) by Lm48; A2: FQ(x) c= the carrier of F_Complex; FQ(x) c= COMPLEX by A2,COMPLFLD:def 1; hence thesis by A1,ZFMISC_1:96; end; theorem Lm57: for x be Element of F_Complex holds the addF of F_Rat = (the addF of FQ_Ring(x))||RAT proof let x be Element of F_Complex; thus the addF of F_Rat = addcomplex|[:RAT,RAT:] by ZFMISC_1:96,RELAT_1:74, VECTSP_1:def 5,RING_3:2,GAUSSINT:13 .= (the addF of FQ_Ring(x))||RAT by Lm56,RELAT_1:74; end; theorem Lm58: for x be Element of F_Complex holds the multF of F_Rat = (the multF of FQ_Ring(x))||RAT proof let x be Element of F_Complex; thus the multF of F_Rat = multcomplex|[:RAT,RAT:] by ZFMISC_1:96,RELAT_1:74, VECTSP_1:def 5,RING_3:3,GAUSSINT:13 .= (the multF of FQ_Ring(x))||RAT by Lm56,RELAT_1:74; end; theorem for x be Element of F_Complex holds F_Rat is Subring of FQ_Ring(x) proof let x be Element of F_Complex; A1: the addF of F_Rat = (the addF of FQ_Ring(x))||the carrier of F_Rat by Lm57; A2: the multF of F_Rat = (the multF of FQ_Ring(x))||the carrier of F_Rat by Lm58; A3: 1.FQ_Ring(x) = 1.F_Complex by Lm52 .= 1.F_Rat by C0SP1:def 3,Th3; 0.FQ_Ring(x) = 0.F_Rat by Lm48,Lm7,SUBSET_1:def 8; hence thesis by Lm55,A1,A2,A3,C0SP1:def 3; end; theorem Th80: for f,g be Element of Polynom-Ring K st f <> 0.Polynom-Ring K & {f}-Ideal is prime & not (g in {f}-Ideal) holds {f,g}-Ideal = the carrier of Polynom-Ring K proof let f,g be Element of Polynom-Ring K; assume that A1: f <> 0.Polynom-Ring K and A2: {f}-Ideal is prime and A4: not g in {f}-Ideal; assume A5: {f,g}-Ideal <> the carrier of Polynom-Ring K; Polynom-Ring K is PID; then consider h be Element of Polynom-Ring K such that A7: {f,g}-Ideal = {h}-Ideal by IDEAL_1:def 27; A8: {f}-Ideal c= {h}-Ideal & {g}-Ideal c= {h}-Ideal by A7,IDEAL_1:69; consider s be Element of Polynom-Ring K such that A9: f = h*s by RING_2:19,A8,GCD_1:def 1; consider t be Element of Polynom-Ring K such that A11:g = h*t by RING_2:19,A8,GCD_1:def 1; f is non zero Element of Polynom-Ring K by A1,STRUCT_0:def 12; then A13:f is prime by A2,RING_2:24; per cases by A9,A13; suppose f divides s; then consider u be Element of Polynom-Ring K such that A16: s = f*u by GCD_1:def 1; A17: f = f*(u*h) by GROUP_1:def 3,A9,A16; reconsider v = u*h as Element of Polynom-Ring K; f * 1.Polynom-Ring K - f*v = 0.Polynom-Ring K by RLVECT_1:5,A17; then f * (1.Polynom-Ring K -v) = 0.Polynom-Ring K by VECTSP_1:11; then 1.Polynom-Ring K + (-v)=0.Polynom-Ring K by A1,VECTSP_2:def 1; then h divides 1.Polynom-Ring K by VECTSP_1:19,GCD_1:def 1; then A27: h is Unit of Polynom-Ring K by GCD_1:def 2; [#] Polynom-Ring K = the carrier of Polynom-Ring K; hence contradiction by A5,A7,A27,RING_2:20; end; suppose f divides h; then consider v be Element of Polynom-Ring K such that A31: h = f*v by GCD_1:def 1; g = f*(v*t) by A11,A31,GROUP_1:def 3; hence contradiction by A4,GCD_1:def 1,RING_2:18; end; end; theorem Th81: for f,g be Element of Polynom-Ring K holds f <> 0.Polynom-Ring K & {f}-Ideal is prime & not g in {f}-Ideal implies {f}-Ideal,{g}-Ideal are_co-prime proof let f,g be Element of Polynom-Ring K; {f,g}-Ideal = {f}-Ideal + {g}-Ideal by IDEAL_1:76; hence thesis by Th80; end; theorem Lm62: for x be Element of F_Complex,a be Element of FQ_Ring(x) ex g be Element of Polynom-Ring F_Rat st a = hom_Ext_eval(x,F_Rat).g proof let x be Element of F_Complex; let a be Element of FQ_Ring(x); ex g1 be object st g1 in dom hom_Ext_eval(x,F_Rat) & a = hom_Ext_eval(x,F_Rat).g1 by FUNCT_1:def 3; hence thesis; end; theorem Th83: for x,a be Element of F_Complex st a <> 0.F_Complex & a in the carrier of FQ_Ring(x) ex g be Element of Polynom-Ring F_Rat st not g in Ann_Poly(x,F_Rat) & a = hom_Ext_eval(x,F_Rat).g proof let x,a be Element of F_Complex; set M = {p where p is Polynomial of F_Rat:Ext_eval(p,x)=0.F_Complex}; assume that A1: a <> 0.F_Complex and A2: a in the carrier of FQ_Ring(x); consider g be Element of Polynom-Ring F_Rat such that A3: a = hom_Ext_eval(x,F_Rat).g by A2,Lm62; take g; thus not g in Ann_Poly(x,F_Rat) proof assume g in Ann_Poly(x,F_Rat); then consider g1 be Polynomial of F_Rat such that A5: g1 = g and A6: Ext_eval(g1,x)=0.F_Complex; thus contradiction by A1,A6,A3,A5,Def9; end; thus thesis by A3; end; theorem Th84: for x,a be Element of F_Complex st x is algebraic & a <> 0.F_Complex & a in the carrier of FQ_Ring(x) ex f,g be Element of Polynom-Ring F_Rat st {f}-Ideal = Ann_Poly(x,F_Rat) & not(g in Ann_Poly(x,F_Rat)) & a = hom_Ext_eval(x,F_Rat).g & {f}-Ideal,{g}-Ideal are_co-prime proof let x,a be Element of F_Complex; assume that A1: x is algebraic and A2: a <> 0.F_Complex and A3: a in the carrier of FQ_Ring(x); consider f be Element of Polynom-Ring F_Rat such that A4: {f}-Ideal = Ann_Poly(x,F_Rat) by Th34,Th3; consider g be Element of Polynom-Ring F_Rat such that A5: not(g in Ann_Poly(x,F_Rat)) and A6: a = hom_Ext_eval(x,F_Rat).g by Th83,A2,A3; A7: {f}-Ideal is prime by A4,A1,Th3,Th39; A8: f <> 0.Polynom-Ring F_Rat by A1,A4,Th35,IDEAL_1:47; {f}-Ideal,{g}-Ideal are_co-prime by A4,A5,A7,A8,Th81; hence thesis by A4,A5,A6; end; theorem Th85: for x,a be Element of F_Complex st x is algebraic & a <> 0.F_Complex & a in the carrier of FQ_Ring(x) holds ex b be Element of F_Complex st b in the carrier of FQ_Ring(x) & a*b = 1.F_Complex proof let x,a be Element of F_Complex; set COPolynomFRat = the carrier of Polynom-Ring F_Rat; set M = {h where h is Polynomial of F_Rat:Ext_eval(h,x)=0.F_Complex}; assume that A1: x is algebraic and A2: a <> 0.F_Complex and A3: a in the carrier of FQ_Ring(x); consider f,g be Element of Polynom-Ring F_Rat such that A4: {f}-Ideal = Ann_Poly(x,F_Rat) and not(g in Ann_Poly(x,F_Rat)) and A6: a = hom_Ext_eval(x,F_Rat).g and A7: {f}-Ideal,{g}-Ideal are_co-prime by A1,A2,A3,Th84; 1.Polynom-Ring F_Rat in {f}-Ideal+{g}-Ideal by A7; then 1.Polynom-Ring F_Rat in {p+q where p,q is Element of Polynom-Ring F_Rat: p in {f}-Ideal & q in {g}-Ideal} by IDEAL_1:def 19; then consider p,q be Element of Polynom-Ring F_Rat such that A10: 1.Polynom-Ring F_Rat = p+q and A11: p in {f}-Ideal and A12: q in {g}-Ideal; A14: {g}-Ideal = the set of all g*s where s is Element of Polynom-Ring F_Rat by IDEAL_1:64; consider s be Element of Polynom-Ring F_Rat such that A15: q = g * s by A12,A14; reconsider p1=p,q1=q, g1=g,s1=s as Polynomial of F_Rat by POLYNOM3:def 10; A16: p+q = p1+q1 by POLYNOM3:def 10; consider p2 be Polynomial of F_Rat such that A17: p2 = p and A18: Ext_eval(p2,x)=0.F_Complex by A4,A11; set b = Ext_eval(s1,x); A20: b = hom_Ext_eval(x,F_Rat).s1 by Def9; A21: dom hom_Ext_eval(x,F_Rat) = the carrier of Polynom-Ring F_Rat by FUNCT_2:def 1; A22: b in the carrier of FQ_Ring(x) by A20,A21,FUNCT_1:def 3; 1.F_Complex = Ext_eval(1_.(F_Rat),x) by Th3,Th18 .= Ext_eval(p1+q1,x) by A10,POLYNOM3:def 10,A16 .= 0.F_Complex + Ext_eval(q1,x) by A17,A18,Th3,Th19 .= Ext_eval(g1 *'s1,x) by A15,POLYNOM3:def 10 .= Ext_eval(g1,x) * Ext_eval(s1,x) by Th3,Th24 .= a*b by A6,Def9; hence thesis by A22; end; theorem for x be Element of F_Complex st x is algebraic holds FQ_Ring(x) is Field proof let x be Element of F_Complex; assume A1: x is algebraic; for a be Element of FQ_Ring(x) st a <> 0.FQ_Ring(x) holds a is left_invertible proof let a be Element of FQ_Ring(x); assume a <> 0.FQ_Ring(x); then A4: a <> 0.F_Complex by SUBSET_1:def 8; a in FQ(x); then reconsider y = a as Element of F_Complex; consider b be Element of F_Complex such that A5: b in the carrier of FQ_Ring(x) and A6: y*b = 1.F_Complex by A1,A4,Th85; reconsider a1=y,b1 = b as Element of FQ_Ring(x) by A5; b1*a1 = 1.F_Complex by A6,Th50 .= 1.FQ_Ring(x) by Lm52; hence thesis; end; then FQ_Ring(x) is almost_left_invertible; hence FQ_Ring(x) is Field; end;