Give an example of an irreducible monic polynomials of degree (a) $4$; (b) $5$ in $\mathbb Z[x]$ that is reducible over $\mathbb Q(\sqrt 2)$, or prove that none exists.
I managed to find a polynomial of degree $4$: $x^4+1=(x^2-\sqrt 2x+1)(x^2+\sqrt 2x+1)$, which is irreducible over $\mathbb Z$ (or equivalently $\mathbb Q$) because it has no rational roots by the rational root test, and cannot have a quadratic factor with rational coefficients because $\mathbb R[x]$ is a UFD, and the two quadratic factors have irrational coefficients.
I've been trying to construct an example for degree $5$, but it seems it's impossible. How can I prove it?