Fields over sums of fractions and isomorphism

Question

Suppose I have, $$ a = (3 + \sqrt{-7})^{1/2} + \frac{1}{(3 + \sqrt{-7})^{1/2}} $$

What can I infer about $\mathbb{Q}(a)$?
And how do I find out if $\mathbb{Q}(a)$ is isomorphic to $\mathbb{Q}[x]/<px^2+qx+r>$ for some given $p,q,r \in \mathbb{Z}$