proof of additive form of Hilbert’s theorem 90

Set n=[L:K].

First, let there be xL such that y=x-σ(x). Then


because x=σn(x).

Now, let Tr(y)=0. Choose zL with Tr(z)0. Then there exists xL with


Since Tr(z)K we have


Now remember that Tr(y)=0. We obtain

(x-σ(x))Tr(z) = yσ(z)+(y+σ(y))σ2(z)++(y+σ(y)++σn-1(y))σn-1(z)
= yTr(z),

so y=x-σ(x), as we wanted to show.

Title proof of additive form of Hilbert’s theorem 90
Canonical name ProofOfAdditiveFormOfHilbertsTheorem90
Date of creation 2013-03-22 15:21:25
Last modified on 2013-03-22 15:21:25
Owner mathwizard (128)
Last modified by mathwizard (128)
Numerical id 4
Author mathwizard (128)
Entry type Proof
Classification msc 12F10
Classification msc 11R32