On the linear independence of algebraic numbers
Louis Joel Mordell · Pacific Journal of Mathematics · 1953
MORDELL 1. Introduction.Besicovitch [l] has proved by elementary methods involving only the concept of the irreducibility of equations the following: THEOREM.Let a i β b lPl> a 2 = ^2 P2» •*• » a s β b s Ps 9where p i9 p 29 ••• , p $ are different primes, and b l9 b 29 ... , b s are positive integers not divisible by any of these primes.If x i9 x 29 9 x s ore positive real roots of the equations and P(xχ, X2' > #s ) * 5 α polynomial with rational coefficients of degree less than or eqv d to n x -1 with respect to x l9 less than or equal to n 2 -1 with respect ί. x 29 and so on 9 then P(xι x 29 ••• , x s ) can vanish only if all its coefficier ts vanish.It ii rather surprising that this has not been proved before, since results of this kind occur as particular cases of a general investigation in the theory of algebraic numbers, and some have been known for many years.We have the well-known general problem:PROBLEM.Let K be an algebraic number field, and letThis holds if either the degrees or the discriminants over K of the fields K(x ί ) 9 K(x 2 ), , K(x s ) are relatively prime in pairs.The first part is a simple consequence of the usual theory of reducibility when s = 2, and the extension is obvious.The second part for s= 2 is given as Theorem 87 in Hubert's report on algebraic number fields, and its proof depends on algebraic number