TITLE: Is there a natural way to view the proof of Hilbert 90?
QUESTION [33 upvotes]: I only know of one proof of Hilbert 90, which is very smart if not magical. See for example http://hilbertthm90.wordpress.com/2008/12/11/hilberts-theorem-90the-math/
Does anyone know of a more intuitive proof or know a good way to view the proof?
I have accepted the answer by Emerton, great thanks as well to David Speyer and Brian Conrad.

REPLY [4 votes]: Another way to conclude that $\tau$ has a fixed point using linear algebra is the following. David Speyer shows that $\tau$ is of order $n$. But, thanks to independence of characters, the minimal polynomial of $\tau$ is $X^n-1$ and thus $\tau$ is a cyclic endomorphism of $L$. Hence 1 is an eigenvalue of $\tau$ with multiplicity one.