Consider the family of monic biquadratic polynomials given by $f\_{a,b}(x) = x^4 + 2ax^2 + b$ with $a,b$ integers. Let $K\_{a,b}$ denote the isomorphism class of quartic fields obtained by adjoining any one of the roots of $f\_{a,b}$. Is there a relatively nice list of criteria to decide whether $K\_{a,b}$ is isomorphic to $K\_{a',b'}$ when $(a,b) \ne (a',b')$? 



Determining when two biquadratic polynomials generate the same field
The necessary and sufficient conditions are as follows:


1) Condition on the common quadratic subfield (as Watson Ladd's answer):
there exists $c\in\Bbb Q$ such that
$${a'}^2-b'=c^2(a^2-b)\;.$$


2) Condition on the discriminant: the polynomial discriminant being $256 b(a^2-b)^2$ we need for some $d\in\Bbb Q$
$$bb'=d^2$$


3) Standard algebraic computation in the field gives the additional
condition for some $e\in\Bbb Q$
$$\dfrac{aa'+c(a^2-b)\pm d}{2}=e^2$$
for a suitable sign $\pm$.


Example: $b=b'=1$, $a=2$, $a'=26$: $c=15$, $d=1$, $e=7$.


