hcju/proofwikips huggingface dataset:
Gauss's Lemma on Primitive Rational Polynomials
Let $\Q$ be the field of rational numbers. Let $\Q \sqbrk X$ be the ring of polynomials over $\Q$ in one indeterminate $X$. Let $\map f X, \map g X \in \Q \sqbrk X$ be primitive polynomials. Then their product $f g$ is also a primitive polynomial.
Preimage of Cover is Cover
Let $\phi: S \to T$ be a mapping between the sets $S$ and $T$. Let $\mathcal U$ be a cover of $T$. Then the set: :$\left\{{\phi^{-1} \left({U}\right): U \in \mathcal U}\right\}$ is a cover of $S$.
Compactness is Preserved under Continuous Surjection
Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ be topological spaces. Let $\phi: T_A \to T_B$ be a continuous surjection. If $T_A$ is compact, then $T_B$ is also compact.
Sigma-Compactness is Preserved under Continuous Surjection
Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ be topological spaces. Let $\phi: T_A \to T_B$ be a continuous surjection. If $T_A$ is $\sigma$-compact, then $T_B$ is also $\sigma$-compact.
Countable Compactness is Preserved under Continuous Surjection
Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ be topological spaces. Let $\phi: T_A \to T_B$ be a continuous surjection. If $T_A$ is countably compact, then $T_B$ is also countably compact.
Sequential Compactness is Preserved under Continuous Surjection
Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ be topological spaces. Let $\phi: T_A \to T_B$ be a continuous surjection. If $T_A$ is sequentially compact, then $T_B$ is also sequentially compact.
Lindelöf Property is Preserved under Continuous Surjection
Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ be topological spaces. Let $\phi: T_A \to T_B$ be a continuous surjection. If $T_A$ is a Lindelöf space, then $T_B$ is also a Lindelöf space.
Multiples of Divisors obey Distributive Law
{{:Euclid:Proposition/VII/6}} In modern algebraic language: :$a = \dfrac m n b, c = \dfrac m n d \implies a + c = \dfrac m n \left({b + d}\right)$
First-Countability is Preserved under Open Continuous Surjection
Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ be topological spaces. Let $\phi: T_A \to T_B$ be a surjective open mapping which is also continuous. If $T_A$ is first-countable, then $T_B$ is also first-countable.
Second-Countability is Preserved under Open Continuous Surjection
Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ be topological spaces. Let $\phi: T_A \to T_B$ be a surjective open mapping which is also continuous. If $T_A$ is second-countable, then $T_B$ is also second-countable.
Arens-Fort Space is Countable
Let $T = \left({S, \tau}\right)$ be the Arens-Fort space. Then $S$ is countably infinite.
Connected iff no Proper Clopen Sets
Let $T = \struct {S, \tau}$ be a topological space. Then $T$ is connected if and only if there exists no proper subset of $S$ which is clopen in $T$.
Element of Principal Ideal Domain is Finite Product of Irreducible Elements
Let $R$ be a principal ideal domain. Let $p \in R$ such that $p \ne 0$ and $p$ is not a unit. Then there exist irreducible elements $p_1, \ldots, p_n$ such that $p = p_1 \cdots p_n$.
Principal Ideal Domain fulfills Ascending Chain Condition
Let $R$ be a principal ideal domain. Then $R$ fulfills the ascending chain condition.
Subspace of Product Space Homeomorphic to Factor Space
Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be a family of topological spaces where $I$ is an arbitrary index set. Let $\displaystyle \struct {X, \tau} = \prod_{i \mathop \in I} \struct {X_i, \tau_i}$ be the product space of $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$. Suppose that $X$ is non-empty. Then for each $i \in I$ there is a subspace $Y_i \subseteq X$ which is homeomorphic to $\struct {X_i, \tau_i}$. Specifically, for any $z \in X$, let: :$Y_i = \set {x \in X: \forall j \in I \setminus \set i: x_j = z_j}$ and let $\upsilon_i$ be the subspace topology of $Y_i$ relative to $\tau$. Then $\struct {Y_i, \upsilon_i}$ is homeomorphic to $\struct {X_i, \tau_i}$, where the homeomorphism is the restriction of the projection $\pr_i$ to $Y_i$.
Closed Set of Countable Fort Space is G-Delta
Let $T = \struct {S, \tau_p}$ be a Fort space on a countably infinite set $S$. Let $H \subseteq S$ be closed in $T$. Then $H$ is a $G_\delta$ set.
Fundamental Theorem of Galois Theory
Let $L / K$ be a finite Galois extension. Let $\operatorname{Gal} \left({L / K}\right)$ denote the Galois group of the extension $L / K$. Let $H$ denote a subgroup of $\operatorname{Gal} \left({L / K}\right)$ and $F$ denote an intermediate field. The mappings: :$H \mapsto L_H$, and :$F \mapsto \operatorname{Gal} \left({L / F}\right)$ are inclusion-reversing and inverses. Moreover, these maps induce a bijection between the normal subgroups of $\operatorname{Gal} \left({L / K}\right)$ and the normal, intermediate extensions of $L / K$.
Ring of Algebraic Integers
Let $K / \Q$ be a number field. Let $\Z \sqbrk x$ denote the polynomial ring in one variable over $\Z$. Let $\mathbb A$ denote the set of all elements of $K / \Q$ which are a root of some monic polynomial $P \in \Z \sqbrk x$. That is, let $\mathbb A$ denote the algebraic integers over $K$. Then $\mathbb A$ is a ring, called the '''Ring of Algebraic Integers'''.
Subtraction of Divisors obeys Distributive Law
{{:Euclid:Proposition/VII/7}} In modern algebraic language: :$a = \dfrac 1 n b, c = \dfrac 1 n d \implies a - c = \dfrac 1 n \paren {b - d}$
Minimal Polynomial is Unique
Let $L / K$ be a field extension and $\alpha \in L$ be algebraic over $K$. Then the minimal polynomial of $\alpha$ over $K$ is unique.
Minimal Polynomial is Irreducible
Let $L / K$ be a field extension. Let $\alpha \in L$ be algebraic over $K$. Then the minimal polynomial in $\alpha$ over $K$ is unique and irreducible.
Equivalence of Definitions of Normal Extension
Let $L / K$ be an algebraic field extension. {{TFAE|def = Normal Extension}}
Abstract Model of Algebraic Extensions
Let $K$ be a field and $\alpha \in \overline{K}$ be an element of the algebraic closure of $K$ which is algebraic over $K$. Let $m_\alpha$ be the minimal polynomial of $\alpha$ over $K$. Then: :$K[\alpha] \cong K[x]/ \langle m_\alpha\rangle$
Union of Local Bases is Basis
Let $T = \left({X, \tau}\right)$ be a topological space. For each $x \in X$, let $\mathcal B_x$ be a local basis at $x$ which consists entirely of open sets. Then $\displaystyle \mathcal B = \bigcup_{x \mathop \in X} \mathcal B_x$ is a basis for the topology $\tau$.
Subtraction of Multiples of Divisors obeys Distributive Law
{{:Euclid:Proposition/VII/8}} In modern algebraic language: :$a = \dfrac m n b, c = \dfrac m n d \implies a - c = \dfrac m n \paren {b - d}$
Proportional Numbers have Proportional Differences
{{:Euclid:Proposition/VII/11}} That is: :$a : b = c : d \implies \left({a - c}\right) : \left({b - d}\right) = a : b$ where $a : b$ denotes the ratio of $a$ to $b$.
Galois Group is Group
Let $L / K$ be a normal extension. Let $\Gal {L / K}$ be the Galois group of $L / K$. Then $\Gal {L / K}$ forms a group under the operation of composition of mappings.
Relation of Ratios to Products
{{:Euclid:Proposition/VII/19}} That is: :$a : b = c : d \iff ad = bc$
Ratios of Fractions in Lowest Terms
Let $a, b, c, d \in \Z_{>0}$ be positive integers. Let $\dfrac a b$ be in canonical form. Let $\dfrac a b = \dfrac c d$. Then: :$a \divides c$ and: :$b \divides d$ where $\divides$ denotes divisibility. {{:Euclid:Proposition/VII/20}}
Gelfond-Schneider Theorem
Let $\alpha$ and $\beta$ be algebraic numbers (possibly complex) such that $\alpha \notin \set {0, 1}$. Let $\beta$ be irrational. Then any value of $\alpha^\beta$ is transcendental.
Associative Algebra has Multiplicative Inverses iff Unitary Division Algebra
Let $\struct {A_R, \oplus}$ be an associative algebra over the ring $A_R$. Then: :$\struct {A_R, \oplus}$ has a unique multiplicative inverse for every non-zero $a \in A_R$ : :$\struct {A_R, \oplus}$ is a unitary division algebra.
Commutator on Algebra is Alternating Bilinear Mapping
Let $\left({A_R, \oplus}\right)$ be an algebra over a ring. Then the commutator on $\left({A_R, \oplus}\right)$ is an alternating bilinear mapping: :$\forall a, b \in A_R: \left[{a, b}\right] = -\left[{b, a}\right]$
Axiom of Subsets Equivalents
The Axiom of Specification states that: :$\forall z: \forall P \left({y}\right): \exists x: \forall y: \left({y \in x \iff \left({y \in z \land P \left({y}\right)}\right)}\right)$ We will prove that this statement is equivalent to the following statements: :$\forall z: \forall A: \left({\left({z \cap A}\right) \in U}\right)$ :$\forall z: \forall A: \left({A \subseteq z \implies A \in U}\right)$ In the above statements, the universe is $U$.
Artin's Theorem on Alternative Algebras
Let $A = \struct {A_R, \oplus}$ be an algebra over the ring $R$ such that $A$ is ''not'' a boolean algebra. Then $A$ is alternative : :$\forall a, b \in A: \paren {a \oplus a} \oplus b = a \oplus \paren {a \oplus b}$ :$\forall a, b \in A: \paren {b \oplus a} \oplus a = b \oplus \paren {a \oplus a}$
Real Numbers form Algebra
The set of real numbers $\R$ forms an algebra over the field of real numbers. This algebra is: :$(1): \quad$ An associative algebra. :$(2): \quad$ A commutative algebra. :$(3): \quad$ A normed division algebra. :$(4): \quad$ A nicely normed $*$-algebra whose $*$ operator is the identity mapping. :$(5): \quad$ A real $*$-algebra.
Real Numbers form Vector Space
The set of real numbers $\R$, with the operations of addition and multiplication, forms a vector space.
Burali-Forti Paradox
The existence of the set of all ordinals leads to a contradiction.
Complex Numbers form Algebra
The set of complex numbers $\C$ forms an algebra over the field of real numbers. This algebra is: :$(1): \quad$ An associative algebra. :$(2): \quad$ A commutative algebra. :$(3): \quad$ A normed division algebra. :$(4): \quad$ A nicely normed $*$-algebra. However, $\C$ is not a real algebra.
Equality implies Substitution
Let $P \left({x}\right)$ denote a Well-Formed Formula which contains $x$ as a free variable. Then the following are tautologies: :$\forall x: \left({P \left({ x }\right) \iff \exists y: \left({y = x \land P \left({y}\right)}\right)}\right)$ :$\forall x: \left({P \left({ x }\right) \iff \forall y: \left({y = x \implies P \left({y}\right)}\right)}\right)$ Note that when $y$ is substituted for $x$ in either formula, it is false in general; compare Confusion of Bound Variables.
Fundamental Law of Universal Class
:$\forall x: x \in \Bbb U$ where: : $\Bbb U$ denotes the universal class : $x$ denotes a set.
Cayley-Dickson Construction forms Star-Algebra
Let $A = \left({A_F, \oplus}\right)$ be a $*$-algebra. Let $A' = \left({A'_F, \oplus'}\right) = \left({A, \oplus}\right)^2$ be the algebra formed from $A$ by the Cayley-Dickson construction. Then $A'$ is also a $*$-algebra.
Empty Set is Small
:$\O \in U$ where $U$ is the universal class.
Algebra from Cayley-Dickson Construction Never Real
Let $A$ be a $*$-algebra. Let $A'$ be constructed from $A$ using the Cayley-Dickson construction. Then $A'$ is not a real algebra.
Cayley-Dickson Construction from Real Algebra is Commutative
Let $A = \left({A_F, \oplus}\right)$ be a $*$-algebra. Let $A' = \left({A_F, \oplus'}\right)$ be constructed from $A$ using the Cayley-Dickson construction. Then $A$ is a real algebra  $A'$ is a commutative algebra.
Cayley-Dickson Construction from Commutative Associative Algebra is Associative
Let $A = \left({A_F, \oplus}\right)$ be a $*$-algebra. Let $A' = \left({A_F, \oplus'}\right)$ be constructed from $A$ using the Cayley-Dickson construction. Then: : $A'$ is an associative algebra : : $A$ is both a commutative algebra and an associative algebra.
Nicely Normed Cayley-Dickson Construction from Associative Algebra is Alternative
Let $A = \left({A_F, \oplus}\right)$ be a $*$-algebra. Let $A' = \left({A_F, \oplus'}\right)$ be constructed from $A$ using the Cayley-Dickson construction. Then $A'$ is a nicely normed alternative algebra  $A$ is a nicely normed associative algebra.
Element of Universe
Let $A$ be a class, which may be either a set or a proper class. Then: :$\forall A: \paren {A \in U \iff \exists x: x = A}$ where $U$ is the universal class.
Conjunction with Tautology
: $p \land \top \dashv \vdash p$
Disjunction with Tautology
: $p \lor \top \dashv \vdash \top$
Conjunction with Contradiction
:$p \land \bot \dashv \vdash \bot$
Disjunction with Contradiction
: $p \lor \bot \dashv \vdash p$
Contradiction is Negation of Tautology
A contradiction implies and is implied by the negation of a tautology: :$\bot \dashv \vdash \neg \top$ That is, a falsehood can not be true, and a non-truth is a falsehood.
Tautology is Negation of Contradiction
A tautology implies and is implied by the negation of a contradiction: :$\top \dashv \vdash \neg \bot$ That is, a truth can not be false, and a non-falsehood must be a truth.
Cayley-Dickson Construction from Nicely Normed Algebra is Nicely Normed
Let $A = \left({A_F, \oplus}\right)$ be a $*$-algebra. Let $A' = \left({A_F, \oplus'}\right)$ be constructed from $A$ using the Cayley-Dickson construction. Then $A'$ is a nicely normed algebra  $A$ is also a nicely normed algebra.
Quaternions form Algebra
The set of quaternions $\Bbb H$ forms an algebra over the field of real numbers. This algebra is: :$(1): \quad$ An associative algebra, but '''not''' a commutative algebra. :$(2): \quad$ A normed division algebra. :$(3): \quad$ A nicely normed $*$-algebra.
Octonions form Algebra
The set of octonions $\Bbb O$ forms an algebra over the field of real numbers. This algebra is: :$(1): \quad$ An alternative algebra, but '''not''' an associative algebra. :$(2): \quad$ A normed division algebra. :$(3): \quad$ A nicely normed $*$-algebra.
Division Algebra has No Zero Divisors
Let $A = \left({A_F, \oplus}\right)$ be an algebra over a field $F$. Then $A$ is a division algebra  it has no zero divisors. That is: :$\forall a, b \in A_F: a \oplus b = \mathbf 0_A \implies a = \mathbf 0_A \lor b = \mathbf 0_A$ If the product of two elements of $A$ is zero, then at least one of those elements must itself be zero. Some sources use this as the definition of a division algebra and from it deduce: :$\forall a, b \in A_F, b \ne \mathbf 0_A: \exists_1 x \in A_F, y \in A_F: a = b \oplus x, a = y \oplus b$
Normed Division Algebra is Division Algebra
Let $A = \struct {A_F, \oplus}$ be a normed divison algebra over a field $F$. Let the unit of $A$ be $1_A$, and the zero of $A$ be $0_A$. Then $A$ is a unitary division algebra. Also: :$\norm {1_A} = 1$ where $\norm {1_A}$ denotes the norm of $1_A$.
Zermelo's Theorem (Set Theory)
Every set of cardinals is well-ordered with respect to $\le$.
Cardinals form Equivalence Classes
Let $\map \Card S$ denote the cardinal of the set $S$. Then $\map \Card S$ induces an equivalence class which contains all sets which have the same cardinality as $S$.
Integers form Commutative Ring with Unity
The integers $\struct {\Z, +, \times}$ form a commutative ring with unity under addition and multiplication.
Ring of Square Matrices over Ring is Ring
Let $R$ be a ring. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\struct {\map {\MM_R} n, +, \times}$ denote the ring of square matrices of order $n$ over $R$. Then $\struct {\map {\MM_R} n, +, \times}$ is a ring.
Ring of Square Matrices over Ring with Unity
Let $R$ be a ring with unity. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\struct {\map {\MM_R} n, +, \times}$ denote the ring of square matrices of order $n$ over $R$. Then $\struct {\map {\MM_R} n, +, \times}$ is a ring with unity.
Ring of Square Matrices over Field is Ring with Unity
Let $F$ be a field. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\struct {\map {\MM_F} n, +, \times}$ denote the ring of square matrices of order $n$ over $F$. Then $\struct {\map {\MM_F} n, +, \times}$ is a ring with unity, but is not a commutative ring.
Matrix Multiplication is not Commutative/Order 2 Square Matrices
Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\map {\MM_R} 2$ denote the $2 \times 2$ matrix space over $R$. The operation of (conventional) matrix multiplication is not commutative over $\map {\MM_R} 2$.
Ring Zero is Unique
Let $\struct {R, +, \circ}$ be a ring. Then the ring zero of $R$ is unique.
Ring Negative is Unique
Let $\struct {R, +, \circ}$ be a ring. Let $a \in R$. Then the ring negative $-a$ of $a$ is unique.
Negative of Ring Negative
Let $\struct {R, +, \circ}$ be a ring. Let $a \in R$ and let $-a$ be the ring negative of $a$. Then: :$-\paren {-a} = a$
Even Perfect Number is Triangular
All perfect numbers which are even are triangular.
Field has no Proper Zero Divisors
Let $\struct {F, +, \times}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Then $\struct {F, +, \times}$ has no proper zero divisors. That is: :$a \times b = 0_F \implies a = 0_F \lor b = 0_F$
Multiplicative Inverse in Field is Unique
Let $\struct {F, +, \times}$ be a field whose zero is $0_F$. Let $a \in F$ such that $a \ne 0_F$. Then the multiplicative inverse $a^{-1}$ of $a$ is unique.
Inverse of Multiplicative Inverse
Let $\struct {F, +, \times}$ be a field whose zero is $0_F$. Let $a \in F$ such that $a \ne 0_F$. Let $a^{-1}$ be the multiplicative inverse of $a$. Then $\paren {a^{-1} }^{-1} = a$.
Integral Multiple Distributes over Ring Addition
Let $\struct {R, +, \times}$ be a ring, or a field. Let $a, b \in R$ and $m, n \in \Z$. Then: :$(1): \quad \paren {m + n} \cdot a = \paren {m \cdot a} + \paren {n \cdot a}$ :$(2): \quad m \cdot \paren {a + b} = \paren {m \cdot a} + \paren {m \cdot b}$ where $m \cdot a$ is as defined in integral multiple.
Integral Multiple of Integral Multiple
Let $\struct {F, +, \times}$ be a field. Let $a \in F$ and $m, n \in \Z$. Then: :$\paren {m n} \cdot a = m \cdot \paren {n \cdot a}$ where $n \cdot a$ is as defined in integral multiple.
Product of Integral Multiples
Let $\struct {F, +, \times}$ be a field. Let $a, b \in F$ and $m, n \in \Z$. Then: :$\paren {m \cdot a} \times \paren {n \cdot b} = \paren {m n} \cdot \paren {a \times b}$ where $m \cdot a$ is as defined in Integral Multiple.
Characteristic of Field by Annihilator
Let $\struct {F, +, \times}$ be a field. Then of the following two cases, exactly one applies:
Annihilator of Ring Always Contains Zero
Let $\struct {R, +, \times}$ be a ring. Let $\map {\mathrm {Ann} } R$ be the annihilator of $R$. Then $0 \in \map {\mathrm {Ann} } R$.
Non-Trivial Annihilator Contains Positive Integer
Let $\left({R, +, \times}\right)$ be a ring with unity. Let $A = \operatorname{Ann} \left({R}\right)$ be the annihilator of $R$. Let $a \in A$ such that $a \ne 0$. Then $A$ contains at least one strictly positive integer.
Area of Sector
Let $\CC = ABC$ be a circle whose center is $A$ and with radii $AB$ and $AC$. Let $BAC$ be the sector of $\CC$ whose angle between $AB$ and $AC$ is $\theta$. :300px Then the area $\AA$ of sector $BAC$ is given by: :$\AA = \dfrac {r^2 \theta} 2$ where: :$r = AB$ is the length of the radius of the circle :$\theta$ is measured in radians.
Rational Numbers form Prime Field
The field of rational numbers $\struct {\Q, +, \times}$ is a prime field. That is, the only subset of $\Q$ which sustains both addition and multiplication are $\Q$ and $\set 0$, and vacuously $\O$.
Field of Integers Modulo Prime is Prime Field
Let $p$ be a prime number. Let $\struct {\Z_p, +, \times}$ be the field of integers modulo $p$. Then $\struct {\Z_p, +, \times}$ is a prime field.
Field has Prime Subfield
Let $\struct {F, +, \times}$ be a field. Then $F$ has a subfield which is a prime field.
Constant Mapping to Identity is Homomorphism/Rings
Let $\struct {R_1, +_1, \circ_1}$ and $\struct {R_2, +_2, \circ_2}$ be rings with zeroes $0_1$ and $0_2$ respectively. Let $\zeta$ be the zero homomorphism from $R_1$ to $R_2$, that is: :$\forall x \in R_1: \map \zeta x = 0_2$ Then $\zeta$ is a ring homomorphism whose image is $\set {0_2}$ and whose kernel is $R_1$.
Constant Mapping to Identity is Homomorphism/Groups
Let $\struct {G_1, \circ_1}$ and $\struct {G_2, \circ_2}$ be groups with identities $e_1$ and $e_2$ respectively. Let $\phi_e: \struct {G_1, \circ_1} \to \struct {G_2, \circ_2}$ be the constant mapping defined as: :$\forall x \in G_1: \map {\phi_e} x = e_2$ Then $\phi_e$ is a group homomorphism whose image is $\set {e_2}$ and whose kernel is $G_1$.
Identity Mapping is Automorphism/Groups
Let $\struct {G, \circ}$ be a group whose identity is $e$. Then $I_G: \struct {G, \circ} \to \struct {G, \circ}$ is a group automorphism. Its kernel is $\set e$.
Identity Mapping is Automorphism/Rings
Let $\struct {R, +, \circ}$ be a ring whose zero is $0$. Then $I_R: \struct {R, +, \circ} \to \struct {R, +, \circ}$ is a ring automorphism. Its kernel is $\set 0$.
Field Homomorphism Preserves Unity
Let $\phi: \struct {F_1, +_1, \times_1} \to \struct {F_2, +_2, \times_2}$ be a field homomorphism. Let: :$1_{F_1}$ be the unity of $F_1$ :$1_{F_2}$ be the unity of $F_2$. Then: :$\map \phi {1_{F_1} } = 1_{F_2}$
Ring Homomorphism from Field is Monomorphism or Zero Homomorphism
Let $\struct {F, +_F, \circ}$ be a field whose zero is $0_F$. Let $\struct {S, +_S, *}$ be a ring whose zero is $0_S$. Let $\phi: F \to S$ be a ring homomorphism. Then either: :$(1): \quad \phi$ is a monomorphism (that is, $\phi$ is injective) or :$(2): \quad \phi$ is the zero homomorphism (that is, $\forall a \in F: \map \phi a = 0_S$).
Zero and Unity of Subfield
Let $\struct {F, +, \times}$ be a field whose zero is $0$ and whose unity is $1$. Let $\struct {K, +, \times}$ be a subfield of $F$.
Galois Field has Non-Zero Characteristic
Let $\GF$ be a Galois field. Then the characteristic of $\GF$ is non-zero.
Power to Characteristic of Field is Monomorphism
Let $F$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let the characteristic of $F$ be $p$ where $p \ne 0$. Let $\phi: F \to F$ be the mapping on $F$ defined as: :$\forall x \in F: \map \phi x = x^p$ Then $\phi$ is a (field) monomorphism.
Scalar Product with Identity
:$\lambda \circ e = 0_R \circ x = e$
Scalar Product with Inverse
:$\lambda \circ \struct {-x} = \struct {-\lambda} \circ x = -\struct {\lambda \circ x}$
Scalar Product with Sum
:$\ds \lambda \circ \paren {\sum_{k \mathop = 1}^m x_k} = \sum_{k \mathop = 1}^m \paren {\lambda \circ x_k}$
Product with Sum of Scalar
:$\ds \paren {\sum_{k \mathop = 1}^m \lambda_k} \circ x = \sum_{k \mathop = 1}^m \paren {\lambda_k \circ x}$
Scalar Product with Product
:$\lambda \circ \paren {n \cdot x} = n \cdot \paren {\lambda \circ x} = \paren {n \cdot \lambda} \circ x$
Expression of Vector as Linear Combination from Basis is Unique
Let $V$ be a vector space of dimension $n$. Let $\mathcal B = \set {\mathbf x_1, \mathbf x_2, \ldots, \mathbf x_n}$ be a basis for $V$. Let $\mathbf x \in V$ be any vector of $V$. Then $\mathbf x$ can be expressed as a unique linear combination of elements of $\mathcal B$. {{explain|The definition we have of linear combination doesn't really work for this. We probably need to expand that definition.}}
Homogeneous Linear Equations with More Unknowns than Equations
Let $\alpha_{ij}$ be elements of a field $F$, where $1 \le i \le m, 1 \le j \le n$. Let $n > m$. Then there exist $x_1, x_2, \ldots, x_n \in F$ not all zero, such that: :$\displaystyle \forall i: 1 \le i \le m: \sum_{j \mathop = 1}^n \alpha_{ij} x_j = 0$ Alternatively, this can be expressed as: If $n > m$, the following system of homogeneous linear equations: {{begin-eqn}} {{eqn | l = 0       | r = \alpha_{11} x_1 + \alpha_{12} x_2 + \cdots + \alpha_{1n} x_n }} {{eqn | l = 0       | r = \alpha_{21} x_1 + \alpha_{22} x_2 + \cdots + \alpha_{2n} x_n }} {{eqn | o = \cdots }} {{eqn | l = 0       | r = \alpha_{m1} x_1 + \alpha_{m2} x_2 + \cdots + \alpha_{mn} x_n }} {{end-eqn}} has at least one solution such that not all of $x_1, \ldots, x_n$ is zero.
Generator of Vector Space Contains Basis
:$G$ contains a basis for $E$.
Cardinality of Linearly Independent Set is No Greater than Dimension
:$H$ has at most $n$ elements.