# About the number of inequivalent irreducible representations of a finite group

We know that if $G$ be a finite group and $F$ be an algebraically closed field whose characteristic does not divide the order of $G$, then the number of inequivalent irreducible $F$-representations of $G$ equals the class number of $G$.

Now, if we suppose that the field $F$ is not necessarily an algebraically closed field but its characteristic does not divide the order of $G$, is it true that the number of inequivalent irreducible $F$-representations of $G$ cannot exceed the class number of $G$?

If the answer is yes, how can we prove that?

I know that each irreducible representation of $G$ over $F$ is completely reducible by Maschke’s Theorem because the characteristic of $F$ does not divide the order of $G$ and $G$ is finite. Also, each representation of $G$ over $F$ can be considered as a representation of $G$ over $\overline{F}$, where $\overline{F}$ is an algebraic closure of $F$. But I have no idea about how they can help.

Yes, the number of irreducible representations cannot exceed the class number (in the case of caprice [autocorrect error. Should be “coprime”] characteristic): Consider the case of an $F$-irreducible module $M$ that becomes reducible over $\bar F$. Then $M$ has an $\bar F$-submodule $N$, and $M$ is simply the direct sum of Galois-conjugates of $N$ (the Galois-sum is an $F$-invariant submodule). That shows that the number of $F$-irreducible representations never exceeds the number of $\bar F$-irreducible representations.
By Maschke’s theorem and the Artin-Wedderburn theorem, the group algebra $k[G]$ decomposes as a finite product $\prod_i M_{n_i}(D_i)$ of matrix algebras over finite-dimensional division algebras $D_i$ over $k$. Here the product runs over all simple modules of $k[G]$.
The center of this product is $\prod_i Z(D_i)$, and hence the number of simple modules is at most the dimension of the center. The center of $k[G]$ has a natural basis given by sums over the conjugacy classes of $G$, and in particular it always has dimension the number of conjugacy classes of $G$.