Show that there are irreducible polynomials of every degree in $\mathbb{Q}[X]$

Question

There is this problem that I would like to ask for any verification whether my answer is correct.

Edited: Thanks @andybenji.

Show that for any $n\ge1$, there exists an irreducible polynomial $f\in\mathbb{Q}[X]$ of degree $n$.

My answer:
For degree n=0, a non-zero constant is a unit, hence it is irreducible in $\mathbb{Q}[X]$.
For all $n\ge1$, $x^n+2$ satisfies Eisenstein's Criterion with p=2, therefore it is irreducible in $\mathbb{Q}[X]$.

I am particularly doubtful about the case of degree 0. Is it correct that a non-zero constant is irreducible in $\mathbb{Q}[X]$? I saw my friend's note which says there are no irreducible polynomials of degree 0. Which one is correct?

Thanks!

Answer 1

As I stated in the comments, the question was unclear, and a possible restatement would be

Show that, for any $n \geq 1$, there exists an irreducible polynomial $f \in \mathbb{Q}[x]$ of degree $n$.

To address the actual question, units are not irreducible. The definition of irreducible states

An element $f \in A$ is called irreducible if $f$ is not zero and not a unit, and for any expression $gh = f$, either $g$ or $h$ is a unit.

So it would, technically, be correct to say there are no irreducible polynomials of degree 0 over a field.