Let $G$ be a finite abelian group and $k$ a field in which the group order is invertible. Then the group algebra $k[G]$ is a semisimple ring by Maschke's theorem. As $k[G]$ is also abelian, the Artin-Wedderburn theorem implies, that $$ k[G]=K_1\oplus\ldots\oplus K_n $$ for fields $K_i$.
I have two questions and I have to admit that they are both a little awkward:
- Is $K_i$ an algebraic field extension of $k$? I am sure that this is true but I can't find an argument.
- If (1) is true and $k$ is algebraically closed, I can deduce, that each irreducible representation of $G$ is one-dimensional. However, this is quite easy to prove directly and the structure of $k[G]$ (obtained by non-trivial theorems) seems to contain more information. Can I deduce something about the number of irreducible representations? What can I say for $k$ not algebraically closed?