Transcendental and Algebraic Numbers
Primarily an advanced study of the modern theory of transcendental and algebraic numbers, this treatment by a distinguished Soviet mathematician focuses on the theory's fundamental methods. The text also chronicles the historical development of the theory's methods and explores the connections with other problems in number theory. The problem of approximating algebraic numbers is also studied as a case in the theory of transcendental numbers.
Topics include the Thue-Siegel theorem, the Hermite-Lindemann theorem on the transcendency of the exponential function, and the work of C. Siegel on the transcendency of the Bessel functions and of the solutions of other differential equations. The final chapter considers the Gelfond-Schneider theorem on the transcendency of alpha to the power beta. Each proof is prefaced by a brief discussion of its scheme, which provides a helpful guide to understanding the proof's progression.
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absolute value algebraic independence algebraic integers algebraic numbers approximation of algebraic arbitrarily small assertion assume common divisor completes the proof congruence conjugate constant different from zero differential equations E-functions edition elliptic function entire function equal to zero fact fi(z field R1 finite algebraic field finite number follows directly function f(z greatest common divisor height H holds infinite irrational irreducible polynomial left member Lindemann theorem linear forms linearly independent Liouville ln H ln q Math measure of transcendence method n+ln H Nauk SSSR nomial nonzero number of solutions number theory obtain the inequality p-adic polynomial of degree polynomial with rational prime ideal problems proved rational field rational fractions rational integers rational integers x1 rational integral coefficients rational numbers respect right member ring of integers roots Russian satisfies the conditions solutions in rational solutions of inequality sufficiently large Suppose system 62 Transcendental Numbers vanish identically variables virtue