I'm writing a paper and want to cite some references to efficiently prove that over any field $k$ of characteristic zero, every irreducible representation of a product of symmetric groups, say

$$ S_{n_1} \times \cdots \times S_{n_p} $$

is isomorphic to a tensor product $\rho_1 \otimes \cdots \otimes \rho_p$ where $\rho_i$ is an irreducible representation of $S_{n_i}$.

I have an open mind about this, but I'm imagining doing it by finding references for these two claims:

If $k$ is an algebraically closed field of characteristic zero, every irreducible representation of a product $G_1 \times G_2$ of finite groups is of the form $\rho_1 \otimes \rho_2$ where $\rho_i$ is an irreducible representation of $G_i$.

If $k$ has characteristic zero and $\overline{k}$ is its algebraic closure, every finite-dimensional representation of $S_n$ over $\overline{k}$ is isomorphic to one of the form $\overline{k} \otimes_k \rho$ where $\rho$ is a representation of $S_n$ over $k$.

Serre's book *Linear Representations of Finite Groups* states the first fact for $k = \mathbb{C}$ but apparently not for a general algebraically closed field of characteristic zero. (It's Theorem 10.) It could be true already for any field of characteristic zero, which would simplify my life.

The second fact should be equivalent to saying that $\mathbb{Q}$ is a splitting field for any symmetric group, which seems to be something everyone knows - yet I haven't found a good reference.

The Great Tome of Representation Theory, which says "$\mathbb{Q}$ is a splitting field for $S_n$", instead of making them wade through a construction and figure out for themselves that $\mathbb{Q}$ is a splitting field for $S_n$. Luckily Corollary 4.16 of Lorenz'sA Tour of Representation Theorysays exactly what I want, right after he does the construction. $\endgroup$1more comment