Is there some algebraic construction/extension to make a reducible polynomial over $\mathbb{Q}$ irreducible?
For example: consider the polynomial $x^4-x^3-x^2+1=(x-1)(x^3-x-1)\in \mathbb{Q}[x]$. Is there some space where this polynomial becomes (or can be easily seen to be) irreducible? And possibly can this construction/extension be expressed in terms of the number fields associated to the roots of the factors?
As a more elementary variant of Joel's answer, you may also consider something like this: Let $f(X)\in\mathbb Z[X]$, and let $a_n$ be the leading coefficient. Then $f_p(X)=f(X)+\frac{1}{p}$ is irreducible for every prime $p$ not dividing $a_n$, because $pf_p(X)$ is the reciprocal of an Eisenstein polynomial.
So irreducible polynomials become arbitrarily close to the polynomial you start with.
The given polynomial $x^4-x^3-x^2+1$ is irreducible in the ring $(x-1){\mathbb Z}[x]$ (the ring of polynomials with integer coefficients that are multiples of $(x-1)$.
I am not sure exactly what you have in mind. But
(1) If your polynomial $P \in \mathbb Q(x)$, and $K%$ is some field containing $\mathbb{Q}$, then obviously it will stay reducible over $K$.
(2) You can assume that your polynomial is monic, and then its coefficients are in $\mathbb Z[1/N]$ for some positive integer $N$, so you can reduce your polynomial mod $p$ for all primes $p$ not dividing $N$. Unfortunately, and obviously, they polynomials you get in $\mathbb F_p[x]$ this way will still be reducible.
So there is no way to make it irreducible by reduction/extension. However, depending on what you want to do after, there is something that you can do:
(3) Put your polynomials $P(x)$ in a family of polynomials $P_t(x)$ depending on some parameter $t \in \mathbb Q$, such that $P_0(X)=P(x)$. It easy to choose the family such that $P_t(x)$ becomes irreducible on $\mathbb Q(x,t)$, and this will imply that $P_t(x)$ will be irreducible over $\mathbb Q(x)$ for all values of $t$ except a (edited:) small set (including your starting point $t=0$) by Hilbert's Irreducibility Theorem (see e.g. wikipedia). Hence, you may be able to prove things for the irreducible polynomials in your family, and deduce similar things by "continuity" for your initial $P_0(X)$.
