Let $A$ be the set of all quadruples $(a\_0,a\_1,a\_2,a\_3) \in {\mathbb Q}^4$ such that
the polynomial $P=X^4+a\_3X^3+a\_2X^2+a\_1X+a\_0$ is irreducible and if $z$ is any root
of $P$, then ${\mathbb Q}(z)$ contains $\sqrt{2}$. Is there a nontrivial polynomial relation
$R(a\_0,a\_1,a\_2,a\_3)=0$ satisified by all $(a\_0,a\_1,a\_2,a\_3) \in A$ ? 



Expressing field inclusions by polynomial equalities on coefficients 
If there was a nontrivial polynomial relation between the coefficients, it would be true for a dense subset (~~reducibility is a nowhere dense condition~~ see comment below) of all polynomials of the form $(x^2+(\alpha +\beta\sqrt{2})x+\gamma+\delta\sqrt{2})(x^2+(\alpha -\beta\sqrt{2})x+\gamma-\delta\sqrt{2})$ with rational $\alpha,\beta,\gamma,\delta$, which would mean the same relation would be true for all real $\alpha,\beta,\gamma,\delta$ as well. But all quartic polynomials are of the form above with real $\alpha,\beta,\gamma,\delta$, so there are no nontrivial relations.


