How to find an element in a field with norm of a specific number? Finding the norm of an element in a field, say, $\mathbb{Z}[\sqrt{19}]$ is rather easy, it just requires the computation of $(a+b\sqrt{19})(a-b\sqrt{19})$. However, given a value, say, 2, how do we find elements in $\mathbb{Z}[\sqrt{19}]$ that have this norm? I guess it boils down to finding integer solutions of the equation $2=a^2-19b^2$, how do we solve this equation?
For some context to my question I'm trying to prove that $\mathbb{Z}[\sqrt{19}]$ is a UFD, and I'm showing that the two $\mathbb{Z}$-primes less than $\sqrt{19}$, 2 and 3, factor into principal ideals, therefore I'm checking that one of ±2 and one of ±3 is a norm.
Thanks in advance!

This is quite difficult in general, but in a quadratic field, knowing the fundamental unit can be useful.
Let $\varepsilon>1$ be the fundamental unit in your quadratic field. If
you seek $\alpha$ with $N(\alpha)=m>0$ then if there is such an $\alpha$
there will be one with $1\le\alpha<\varepsilon$. Its conjugate $\alpha^*$
satisfies $m\ge\alpha^*>m/\varepsilon$. (If $m>0$ we need slightly different inequalities). This gives bounds on $\alpha\pm\alpha^*$ which must be
an integer or $\sqrt d$ times an integer (where your field is $\Bbb Q(\sqrt d)$). This gives a finite (maybe large) set of possibilities to check.
