Irreducibility of $f(x)=x^4+4x-1$

Question

Is the polynomial $f(x)=x^4+4x-1$ irreducible over $\mathbb{Q}(\sqrt{7}i)$.

Actually I can't find any way to do this. Any help/hint in this regards would be highly appreciated. Thanks in advance!

Answer 1

Note the polynomial is primitive (constant term $1$), it suffices to prove irreducibility in the ring of integers $\mathcal{O}$ of $\mathbb{Q}(\sqrt{-7})$.

If $r\in \mathbb{Q}(\sqrt{-7})$ such that $f(r)=0$, then $r$ must be a unit of $\mathcal{O}$, so $r=\pm 1$, these are quickly rule out. For expansion like $$x^4+4x-1=(x^2+ax+b)(x^2-ax+c)$$ where $a,b,c\in \mathcal{O}$, this is equivalent to $$bc = -1 \qquad b+c = a^2 \qquad a(c-b) = 4$$ Hence $b,c$ are units. Thus $b=1,c=-1$ or $b=-1,c=1$, in both cases $b+c =0$, hence $a=0$, but this contradicts to the third equation.