Abelian implies every irreducible representation is one-dimensional

Statement
For an abelian group, any irreducible representation over a splitting field is one-dimensional.

In particular, for a finite abelian group of order $$n$$, the fact about::degrees of irreducible representations over a splitting field are $$1,1,1,\dots,1$$ and there are $$n$$ of them.

Related specific information

 * Linear representation theory of cyclic groups
 * Linear representation theory of finite abelian groups

Generalizations

 * Degree of irreducible representation divides index of abelian normal subgroup
 * Degree of irreducible representation is bounded by index of abelian subgroup
 * Number of one-dimensional representations equals order of abelianization