The Error of His Ways

Question

Problem: A student claims that x-1 is not irreducible because $x-1 = (\sqrt{x} -1)(\sqrt{x} + 1)$. Explain the error of his ways.

Definition: An element $p$ in a domain $R$ is irreducible if $p$ is neither $0$ nor a unit and, in every factorization $p=uv$, either $u$ or $v$ is a unit.

It's hard to say what's wrong. The problem doesn't specify how we are to interpret the symbols. Are we working $\Bbb{Z}[x]$? $\Bbb{Q}[x]$? $\Bbb{R}[x]$? Or perhaps $\Bbb{Z}_n[x]$? Does the symbol $\sqrt{x}$ even make sense in any of those settings?

Answer 1

Even if we're working in a ring where $\sqrt x$ makes sense, the student's argument will only work if he also argues that neither $\sqrt x - 1$ nor $\sqrt x + 1$ is a unit in that ring.

Since we don't know what the ring is, it's possible that he can in fact do that (that would be the case if the ring is $\mathbb R[\sqrt x]$, for example) -- but it is still his error that he is not providing that argument.

Answer 2

If we are indeed working in a polynomial ring, then I think you are correct that the symbol $\sqrt{x}$ makes no sense in this setting.

More generally (as you pointed out), what do we mean when we say that a polynomial of degree greater than one is irreducible? It may be reducible over $\mathbf R$ but not over $\mathbf Q$. And it'll always be reducible over $\mathbf C$ by the Fundamental Theorem of Algebra.