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?

Well, there are two obvious strategies: show that the ideal $(q,\sqrt{t})$ is maximal, or directly exhibit inverses in the quotient ring. Have you tried to do either of these? What difficulties did you encounter? — Alex Kruckman, May 03 '19 at 21:59
I use the second method by showing each nonzero element in the quotient ring has a unit. But encounter some difficult to find the unit. — Jonny, May 03 '19 at 22:03
Ok, how do you characterize the nonzero elements of the quotient ring? You'll get better responses on this site if you include detailed descriptions of what you've tried in your question. — Alex Kruckman, May 03 '19 at 22:07
I take an arbitrary nonzero element y = a + ( q, $\sqrt{t}$ ) in the quotient ring. Let's say a = c + d$\sqrt{t}$. Then WTS $\exists$ z = b + ( q, $\sqrt{t}$ ) such that zy = ab + ( q, $\sqrt{t}$ ) = 1+ ( q, $\sqrt{t}$ ). Then it suffices to show ab - 1 $\in$ ( q, $\sqrt{t}$ ). i.e. ( c + d$\sqrt{t}$)( e + f$\sqrt{t}$ ) $-$ 1 = qw + $\sqrt{t}$v for some w,v $\in$ $\mathbb{Z}$[$\sqrt{t}$]. But I feel difficult to find integers e and f to make this equation hold. — Jonny, May 03 '19 at 22:21
$!\bmod q,\sqrt t!:\ , q\equiv 0\equiv\sqrt t ,\Rightarrow\ a + b\sqrt t ,\equiv, a\bmod q\ \ \ $ — Bill Dubuque, May 03 '19 at 22:42
By the prior comment, every element in your quotient ring is equivalent to an integer; and since also $q = 0$ your quotient ring is an image of $,F=\Bbb Z/q,$, necessarily $F$ since $1\not\in(q,\sqrt t).\ \ $ — Bill Dubuque, May 03 '19 at 23:08

score 2 · Accepted Answer · answered May 03 '19 at 23:26

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.

show a quotient ring is a field

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