Factor $X^7-(4+i)\in\mathbb{Q}(i)[X]$...if possible.

Question

I think $X^7-(4+i)\in\mathbb{Q}(i)[X]$ is irreducible (simply because I don't know how to go about factoring it). Would it suffice to show that it is irreducible over $\mathbb{Z}[i]$?

If so, I can consider \begin{align*} X^7-(4+i)+\langle i\rangle=X^7-1+\langle i\rangle\in\dfrac{\mathbb{Z}[i]}{\langle i\rangle}[X]\cong\mathbb{Z}[X], \end{align*} where $\langle i\rangle$ is the prime ideal (since $\mathbb{Z}[i]/\langle i \rangle\cong\mathbb{Z}$ is a domain) generated by $i$. Am I completely off my rocker?

Answer 1

Hint: I think you might be expected to notice that $4+i$ is prime in $\mathbb Z[i]$ and to think about whether you can adapt Eisenstein's Criterion to this slightly different case.

Answer 2

Let $R$ be a GCD domain and $K$ its field of fractions. A non-constant polynomial $f\in R[X]$ is irreducible if and only if it is primitive and irreducible in $K[X]$; see here.

Fortunately $\mathbb Z[i]$ is a GCD domain (it's even a Euclidean domain), so $X^7-(4+i)$ is irreducible in $\mathbb Q(i)[X]$ iff $X^7-(4+i)$ is irreducible in $\mathbb Z[i][X]$. Now you can use the Eisenstein's criterion for the prime ideal $\mathfrak p=(4+i)$. (Why is this a prime ideal? Well, this is proved e.g. here: $a+bi$ is prime in $\mathbb{Z}[i]$ if and only if $a^2+b^2$ is prime in $\mathbb{Z}$. In fact, $\mathbb Z[i]/(4+i)\simeq\mathbb Z/17\mathbb Z$ which is a finite field.)