Let $X\subseteq \mathbb{C}^n$ be an irreducible variety defined over $\mathbb{Q}$. I would like to show that for all but finitely many prime $p$ the variety $X(\mathbb{F}\_p)$, defined over $\mathbb{F}\_p$, is geometrically irreducible, i.e., $X(\overline{\mathbb{F}}\_p)$ is irreducible. Can someone give me a hint or a reference?


Thanks



Geometrically irreducible variety
I agree with the OP that there is no need to use fancy and sophisticated language and tools. (Maybe what he calls *variety* is nowadays better called *algebraic set*.)


(a) The case that $X$ is a hypersurface is well known, some people call it the Bertini-Noether Theorem. Here one has to show that an absolutely irreducible polynomial $f(\bf x)\in{\mathbb Q}[\bf x]$ remains absolutely irreducible modulo all but finitely many primes. That follows from a straightforward application of Hilbert's Nullstellensatz, see e.g. Chapter VIII, $\S$5, Prop. 7 (page 157) of Lang's *Diophantine Geometry* or
Chapter IX, $\S$5, Prop. 5.3 (page 241) of Lang's *Fundamentals of Diophantine Geometry*.


(b) The general case can be reduced to the case of a hypersurface because $X$ is birationally equivalent to a hypersurface (this uses the primitive element theorem). Indeed, a generalization of the OP's question is Prop. 10.4.2 in the second and third edition of *Field Arithmetic* by Fried and Jarden.


