Baker's theorem
In transcendental number theory, a mathematical discipline, Baker's theorem gives a lower bound for the absolute value of linear combinations of logarithms of algebraic numbers. The result, proved by , subsumed many earlier results in transcendental number theory and solved a problem posed by Alexander Gelfond nearly fifteen years earlier. Baker used this to prove the transcendence of many numbers, to derive effective bounds for the solutions of some Diophantine equations, and to solve the class number problem of finding all imaginary quadratic fields with class number 1. History. To simplify notation, let formula_1 be the set  of logarithms to the base "e" of nonzero algebraic numbers, that is where formula_3 denotes the set of complex numbers and formula_4 denotes the algebraic numbers (the algebraic completion of the rational numbers formula_5). Using this notation, several results in transcendental number theory become much easier to state. For example the Hermite–Lindemann theorem becomes the statement that any nonzero element of formula_1 is transcendental. In 1934, Alexander Gelfond and Theodor Schneider independently proved the Gelfond–Schneider theorem. This result is usually stated as: if "a" is algebraic and not equal to 0 or 1, and if "b" is algebraic  and irrational, then "a" is transcendental. Note that this includes all determinations of "a", which in most cases constitutes infinitely many numbers. Equivalently, though, it says that if formula_7 are linearly independent over the rational numbers, then they are linearly independent over the algebraic numbers. So if formula_7 and λ isn't zero, then the quotient λ/λ is either a rational number or transcendental. It cannot be an algebraic irrational number like . Although proving this result of "rational linear independence implies algebraic linear independence" for two elements of formula_1 was sufficient for his and Schneider's result, Gelfond  felt that it was crucial to extend this result to arbitrarily many elements of formula_10 Indeed, from : This problem was solved fourteen years later by Alan Baker and has since had numerous applications not only to transcendence theory but in algebraic number theory and the study of Diophantine equations as well. Baker received the Fields medal in 1970 for both this work and his applications of it to Diophantine equations. Statement. With the above notation, Baker's theorem is a nonhomogeneous generalization of the Gelfond–Schneider theorem. Specifically it states: Just as the Gelfond–Schneider theorem is equivalent to the  statement about the transcendence of numbers of the form "a", so too Baker's theorem implies the transcendence of numbers of the form where the "b" are all algebraic, irrational, and 1, "b", …, "b" are linearly independent over the rationals, and the "a" are all algebraic and not 0 or 1. is either 0 or satisfies where and the field generated by formula_19 and formula_20 over the rationals has degree at most "d". In the special case when "β" = 0 and all the formula_14 are rational integers, the rightmost term log Ω can be deleted. An explicit result by  Baker and Wüstholz for a linear form Λ with integer coefficients yields a lower bound of the form where and "d" is the degree of the number field generated by the formula_24 Baker's method. Baker's proof of his theorem is an extension of the argument given by . The main ideas of the proof are illustrated by the proof of the following qualitative version of the theorem of described by : The precise quantitative version of Baker's theory can be proved by replacing the conditions that things are zero by conditions that things are sufficiently small throughout the proof . The main idea of Baker's proof is to construct an auxiliary function formula_27 of several variables that vanishes to high order at many points of the form formula_28 then repeatedly show that it vanishes to lower order at even more points of this form. Finally the fact that it vanishes (to order 1) at enough points of this form implies using Vandermonde determinants that there is a multiplicative relation between the numbers "a". Construction of the auxiliary function. Assume there is a relation for algebraic numbers "α", …, "α", "β", …, "β". The function Φ is of the form The integer  coefficients "p" are chosen so that they are not all zero and Φ and its derivatives of order at most some constant "M" vanish at formula_28 for integers formula_32 with formula_33 for some constant "h". This is possible because these conditions are homogeneous linear equations in the coefficients "p", which have a non-zero solution provided the number of unknown variables "p" is larger than the number of equations. The linear relation between the logs of the α's is needed to cut down the number of linear equations that have to be satisfied. Moreover, using Siegel's lemma, the  sizes of the coefficients "p" can be chosen to be not too large. The constants "L", "h", and "M" have to be carefully adjusted so that the next part of the proof works, and are subject to some constraints, which are roughly: The constraints can be satisfied by taking "h" to be sufficiently large, "M" to be some fixed power of "h", and "L" to be a slightly smaller power of "h". Baker took "M" to be about "h" and "L" to be about "h". The linear relation between the logarithms of the α's is used to reduce "L  slightly; roughly speaking, without it the condition L" must be larger than about "M""h" would become "L" must be larger than about "Mh", which is incompatible with the condition that "L" is somewhat smaller than "M". Zeros of the auxiliary function. The next step is to show that Φ vanishes to slightly smaller order at many more points of the form formula_34 for integers "l". This idea was Baker's key innovation: previous work on this problem involved trying to increase the number of derivatives that vanish while keeping the number of points fixed, which does not seem to work  in the multivariable case. This is done by combining two ideas; First one shows that the derivatives at these points are quite small, by using the fact that many derivatives of Φ vanish at many nearby points. Then one shows that derivatives of Φ at this point are given by algebraic integers times known constants. If an algebraic integer has all its conjugates bounded by a known constant, then it cannot be too small unless it is zero, because the product of all conjugates of a nonzero algebraic integer is at least 1 in absolute value. Combining these two  ideas implies that Φ vanishes to slightly smaller order at many more points formula_35 This part of the argument requires that Φ does not increase too rapidly; the growth of Φ depends on the size of "L", so requires a bound on the size of "L", which turns out to be roughly that "L" must be somewhat smaller than "M". More precisely, Baker showed that since Φ vanishes to order "M" at "h" consecutive integers, it also vanishes to order "M"/2 at "h" consecutive integers 1, 2, 3, …. Repeating this argument "J" times shows that Φ vanishes to order "M"/2  at "h" points, provided that "h" is sufficiently large and "L" is somewhat smaller than "M"/2. One then takes "J" large enough that: ("J" larger than about 16"n" will do if "h" > "L") so that: Completion of the proof. By definition formula_38 can be written as: Therefore as "l" varies we have a system of ("L" + 1) homogeneous linear equations in the ("L" + 1) unknowns which by assumption has a non-zero solution, which in turn implies the determinant of the matrix of coefficients must vanish. However this matrix is a Vandermonde matrix and the formula for  the determinant of such a matrix forces an equality between two of the values: so formula_41 are multiplicatively dependent. Taking logs shows that formula_42 are linearly dependent over the rationals. Extensions and generalizations. in fact gave a quantitative version of the theorem, giving effective lower bounds for the linear form in logarithms. This is done by a similar argument, except statements about something being zero are replaced by statements giving a small upper bound for it, and so on. is nonzero for nonzero algebraic numbers "β", …, "β", "α", …, "α", and moreover giving an effective lower bound for it. The proof  is similar to the homogeneous case: one can assume that and one inserts an extra variable "z" into Φ as follows: Corollaries. As mentioned above, the theorem includes numerous earlier transcendence results concerning the exponential function, such as the Hermite–Lindemann theorem and Gelfond–Schneider theorem. It is not quite as encompassing as the still unproven Schanuel's conjecture, and does not imply the six exponentials theorem nor, clearly, the still open four exponentials conjecture. The main reason Gelfond desired an extension of his result was not just for a slew of new transcendental numbers. In 1935 he used the  tools he had developed to prove the Gelfond–Schneider theorem to derive a lower bound for the quantity where "β" and "β" are algebraic and "λ" and "λ" are in formula_1. Baker's proof gave lower bounds for quantities like the above but with arbitrarily many terms, and he could use these bounds to develop effective means of tackling Diophantine equations and to solve Gauss' class number problem. Extensions. Baker's theorem grants us the linear independence over the algebraic numbers of logarithms of algebraic numbers. This is weaker than proving their "algebraic" independence. So far no progress has been  made on this problem at all. It has been conjectured that if "λ", …, "λ" are elements of formula_1 that are linearly independent over the rational numbers, then they are algebraically independent too. This is a special case of Schanuel's conjecture, but so far it remains to be proved that there even exist two algebraic numbers whose logarithms are algebraically independent. Indeed, Baker's theorem rules out linear relations between logarithms of algebraic numbers unless there are trivial reasons for them; the next most simple case, that of ruling out homogeneous quadratic relations, is the still open four exponentials conjecture . Similarly, extending the result to algebraic independence but in the p-adic setting, and using the "p"-adic logarithm function, remains an open problem. It is known that proving algebraic independence of linearly independent "p"-adic logarithms of algebraic "p"-adic numbers would prove Leopoldt's conjecture on the "p"-adic ranks of units of a number field.