## Weierstrass’ proof of the Lindemann-Weierstrass Theorem (Part 1 of 3)

In part 1 of his 1885 paper, Weierstrass demonstrated the following lemma, which is both a simplification and a generalisation of the method developed by Hermite to prove the transcendence of e:

Lemma:  Let f(z) be a polynomial of degree n+1 with integer coefficients and whose roots z0, … , zn are all distinct.  Then there exists a system of polynomials g0(z), … , gn(z) of degree not greater than n in z, and with integer coefficients, such that (i) each of the differences

$g_{\nu}(z_0)e^{z_\lambda}-g_{\nu}(z_{\lambda})e^{z_0}$

(where ν,λ can take any of the values 0, 1, … , n) can be made arbitrarily small in absolute value, and (ii) the determinant whose elements are gν(zλ) is non-zero.

In the attached pdf (updated on 1 July 2013), I reproduce Weierstrass’ proof of this lemma, with some minor modifications.