Let $G$ and $H$ by finite groups. Let $V$ and $U$ be irreducible representations of $G$ and $H$, respectively.
When the ground field is $\mathbb{C}$, I know how to show that $V\otimes U$ is an irreducible representation of $G\times H$ (using character theory).
What about other ground fields? Is $V\otimes U$ guaranteed to be an irreducible rep of $G\times H$ if the ground field is $F$ and $\operatorname{char}F\nmid|G|$?
No. For example, let $G$ be cyclic of order $3$. Then $G$ has an ireducible module $U$ of dimension $2$ over ${\mathbb Q}$. But $U \otimes U$ is not irreducible for $G \times G$; it decomposes into two $2$-dimensional modules.