Archive for February, 2010

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

February 27, 2010

In part 3 of his 1885 paper, Weierstrass proved the theorem, which in the form stated by him is: if each of z1, … , zn is algebraic and distinct, and A1, … , An are algebraic then

A_1e^{z_1}+ ... + A_ne^{z_n}

cannot be zero unless all of A1, … , An are zero.

In the attached pdf (revised on 6 Aug 2013 – I am grateful to Fang Sun for pointing out a problem with the original which is addressed in this revision), I reproduce part 3 of Weierstrass’ paper, with some minor modifications and some added explanation.

Weierstrass’ proof of the Lindemann-Weierstrass Theorem (Part 2 of 3)

February 16, 2010

In part 2 of his 1885 paper, Weierstrass reproduced Lindemann’s proof that if z is algebraic,


is not zero.

Weierstrass’ proof of Lindemann’s result is concise and elegant, while still displaying its origins in Hermite’s ground-breaking proof that e is transcendental.  This contrasts with some of the ‘simplifications’ that have followed, for example the surprisingly cumbersome treatment given by Hardy and Wright.

In the attached pdf (updated 25 July 2013), I reproduce part 2 of Weierstrass’ paper, with some minor modifications and some added material on symmetric polynomials.  The latter draws on the description of Waring’s method in Tignol’s book ‘Galois’ Theory of Algebraic Equations’.

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

February 13, 2010

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


(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.

Lindemann-Weierstrass Theorem

February 6, 2010

Lindemann showed that pi is transcendental by showing that if the complex number z is algebraic (the root of an integral polynomial) then ez +1 cannot be zero.

Lindemann also claimed, and Weierstrass proved rigorously, that this result can be generalised to the following: if each of z1, … , zn is algebraic, and N1, … , Nn are algebraic and non-zero, then

N_1e^{z_1}+ ... +N_ne^{z_n}

cannot be zero.

It follows immediately that if z is algebraic and non-zero, ez, sin z and cos z are transcendental and corresponding results can be obtained for the inverse functions.

Although proofs of the transcendence of pi are found in many texts, for example the proof in Alan Baker’s book ‘Transcendental Number Theory’  discussed in the previous post, full proofs of the Lindemann-Weierstrass Theorem are harder to find.  Baker provides a two-page proof, which is widely cited, but it is extremely condensed and key steps are omitted.  I have been forced to go back to Weierstrass’ original 1885 paper to find a full proof.

Weierstrass’ paper was published in the rather fulsomely titled ‘Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin’, pages 1067 to 1085.  A link to the journal is provided on the ‘Journals’ page.

Weierstrass’ paper is in three parts, and in the next three posts I will discuss each of these parts in turn.