nt.number theory - Expressing field inclusions by polynomial equalities on coefficients

This may ramble a bit much, but I hope it provides some help in how to think about the problem.

Let's see what your extension of fields looks like. We have 4 possible extensions (perhaps the same) So that any of them is

$mathbb Q(z_i)$

$|$

$mathbb Qleft(sqrt2right)$

$|$

$mathbb Q$

Where $z_i$ ranges of the 4 possible roots $z_1,...,z_4.$ Then $mathbb Q(z_1)$ is degree 4 (since the polynomial is irreducible), but this polynomial factors into a product of quadratics over $mathbb Qleft(sqrt2right).$ So indeed we've reduced to having only two possible extensions, in that the two roots of the same quadratic generate the same extension over $mathbb Q(sqrt2).$

However, except for this restriction, I don't see anything else to lead to a relation on the coefficients. Hopefully this will help you or someone else get a start on the problem.

One further thought:

Since the roots appear in pairs (say $z_1$ and $z_2$ are conjugate over $mathbb Qleft(sqrt 2right)$) then one can generate $sqrt 2$ with either pair, and subtract them. However, I don't immediately see a way to gather that information from the symmetric polynomials of the roots (a.k.a. the coefficients $a_1, ldots, a_4.$)