**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 e*^{α} 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 2*πi* be; but *e*^{2}^{πi} = 1. Similarly *e*^{log 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