Hilbert’s Theorem 90
Posted: May 24, 2024 in Basic Algebra, Fields Tags: automorphism group, Hilbert’s Theorem 90
In this post, which is one of the oldest posts in this blog, we proved a special case of an important and interesting result in field theory known as Hilbert’s Theorem 90; that is, we showed that if is a field and is an automorphism of of order two, then is the set of all solutions of the equation We now prove the general case of the theorem.
Hilbert’s Theorem 90. Let be a field, and an automorphism of of order Then the set of all solutions of the equation is
Proof. If for some then
because for all and because the order of is Conversely, suppose that for some Clearly Since the order of is the functions where is the identity map, are pairwise distinct and hence, by this post, they are -linearly independent. Thus
and so there exists such that Then, since
we get that
because for and also for because So
and hence where
