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Auxiliary Polynomials in Number Theory by David Masser

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13

Transcendence II – Hermite–Lindemann

By now we have stockpiled so many weapons of mass destruction that the transcendence of e, first proved by Hermite in an amazing work no less astonishing for the fact of his age 51, is a relatively defenceless target. We can even prove the following

Theorem 13.1 Suppose α ≠ 0 is algebraic in C. Then eα is transcendental.

This includes the transcendence of π (up to now not proved even to be irrational in these pages), because if it were algebraic then so would 2πi be; but e2πi = 1. Similarly elog 2 = 2 so we get the transcendence of log 2. More generally we deduce the transcendence of any non-zero determination of log β for any non-zero algebraic β.

The above theorem has the same shape as Theorems 11.2

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