Does a multivariate polynomial stay irreducible in transcendental field extension?

Question

If $F$ is an algebraically closed field, and $p \in F[X_1, ..., X_n]$ irreducible, and $K$ a (transcendental) field extension of $F$, is $p$ always irreducible in $K$?

Answer 1

Let $X = \mathrm{Spec}\, F[X_1,...,X_n]/p$. Then $X$ is an irreducible scheme over a separably (even algebraically) closed field $F$. Now the following result holds.

Lemma 33.8.3 in stacks project Part 2 Let $X$ be an scheme over a separably closed field $F$. If $X$ is irreducible, then for every field extension $F\subseteq K$ scheme $X_K = \mathrm{Spec\, K}\times_{\mathrm{Spec}\,F}X$ is irreducible.

Since in our case $X_K=\mathrm{Spec}\, K[X_1,...,X_n]/p$, we derive that $p$ is irreducible in $K[X_1,...,X_n]$.