Decomposition of an algebraic set over a number field, into absolutely irreducible varieties, possibly in another extension

Question

let $f \in \mathbb{Z}[x_{1}, \ldots , x_{m}]$ be a multivariable polynomial and $V$ be the algebraic set defined over $\mathbb{Q}$, which might have dimension zero or might be empty. The claim is following:

There exists a normal extension $N$ over $\mathbb{Q}$ such that $V = \cup_{i=1}^{n} V_{i}$, where each of the $V_{i}$ are absolutely irreducible varieties over $N$.

I am trying to self-study and I have so many questions about this.

1. Does the normal extension $N$, whose existence we need to prove, a finite extension? Especially when $f$ is of $\geq 2$ variables?
2. I have no intuition on varieties over fields that are not algebraically closed. It appears that $N$ would not necessarily be algebraically closed.

If $f$ was of one-variable, we could factor $f = f_{1} \cdots f_{\nu}$ where each of $f_{1}, \ldots, f_{\nu}$ are irreducible. Then I could take a root $\gamma_{i}$ of each $f_{i}$ and perhaps $N = \mathbb{Q}(\gamma_{1}, \ldots , \gamma_{\nu}).$ I am not 100% sure even about this case because I still need to show that varieties generated by $f_{i}$ are absolutely irreducible over $N$. When there are more than $1$ variable, I am totally stumped.

Answer 1

Let me first point out that you can always take $N=\overline{\Bbb Q}$: as any irreducible variety over an algebraically closed field is geometrically/absolutely irreducible, this will always suffice.

One can also find a finite extension $N$ of $\Bbb Q$ where this behavior can be seen, too: given an ideal $I\subset\Bbb Q[x_1,\cdots,x_n]$, take a primary decomposition of $I\otimes_{\Bbb Q} \overline{\Bbb Q}$ inside $\overline{\Bbb Q}[x_1,\cdots,x_n]$: this gives us ideals $\mathfrak{q}_i$ which are the ideals of the geometrically irreducible components $V_i$ of $V$ after base change to the algebraic closure. Now take a finite list of generators for each ideal, consider the finite extension of $\Bbb Q$ obtained by adjoining all of the coefficients of those generators, and take the normal closure as $N$.

As far as intuition for varieties over non-algebraically closed fields, I highly recommend upgrading to schemes - this is one of the major problems they were invented to deal with, after all. Once you do this, there's a lot of fun stuff to say. The first, most pictorial idea I can relate here is that a variety over a field $k$ naturally "unfolds" to a variety over $\overline{k}$: see here, for instance.