TITLE: Brauer group of a finite field is trivial
QUESTION [5 upvotes]: I know that all finite skew fields are field. How does it follow from this fact that the Brauer group of a finite field is trivial?

REPLY [6 votes]: The Brauer group of a field $k$ is the group of (isomorphism classes of) division algebras whose center is $k$. Since any finite division algebra is a field, and hence is its own center, the only division algebra whose center is $\mathbb{F}_q$ is $\mathbb{F}_q$ itself. The statement that every finite division algebra is a field is known as Wedderburn's Theorem. (more detail here: http://en.wikipedia.org/wiki/Wedderburn%27s_little_theorem)
You may be more familiar with the definition of Brauer group as a group of classes of central simple algebras over $k$, but notice that in this definition each equivalence class contains exactly one division algebra.