Galois theory: Hilbert's theorem 90

This lecture is part of an online graduate course on Galois theory.

We discuss two forms of Hilbert's theorem 90: the original version for cyclic extensions, and Noether's more general version for arbitrary finite Galois extensions. The proofs use a lemma of Artin about the linear independence of group characters.

[portport]

The last minute of every video is always a gold mine

[alexeystaroletov]

It seems that at some point (since 9:00) arbitrary group G becomes equal to M^ast.
To fix the proof one need to consider b such that sigma_2(b)\neq1. Then we have two equalities:
and for all c\in M^\ast
But
After subtracting equalities, we find that -
this is a shorter relation on characters and it is nontrivial since a_2\neq a_2*sigma_2*(b) ; a contradiction.

This is very similar to proving that the eigenvectors associated with distinct eigenvalues are linearly independent.

[xcl]

I thought Hilbert try to type Hilbert's theorem () but forget to press Shift

[MudithaMaths]

@17:10: Shouldn't it read M = K(beta)?

[yt-]

I saw Artin's Lemma in some book as Dedekind's

[willnewman]

18:10 "we have already shown this for arbitrary separable extensions"
What video was this in?

[lagrangiangrassmannian]

Maybe he messed up the equations about 11:00