show a quotient ring is a field

Question

let q be a prime number which divides integer t,
how can we show quotient ring $\mathbb{Z}$[$\sqrt{t}$]/(q, $\sqrt{t}$) is a field?

Answer 1

Every element of $\mathbb{Z}[\sqrt{t}]$ has the form $x=a+b\sqrt{t}$ for uniquely determined integers $a,b\in \mathbb{Z}$. So define a map $\varphi:\mathbb{Z}[\sqrt{t}]\rightarrow \mathbb{Z}/q\mathbb{Z}$ by $$x\mapsto a\pmod{q}.$$

Check that this is surjective and $\ker(\varphi)=(q,\sqrt{t})$. Now think about the first isomorphism theorem.