# Math Help - Quotient Fields

1. ## Quotient Fields

The problem states:

Determine Q(D) for D = {m + n√2 | m,n in Z} where D is an intefral domain and Q(D) its quotient field.

I know that the answer is Q(D) Q(√2), but am not sure how to come to this conclusion.

2. Originally Posted by page929
The problem states:

Determine Q(D) for D = {m + n√2 | m,n in Z} where D is an intefral domain and Q(D) its quotient field.

I know that the answer is Q(D) Q(√2), but am not sure how to come to this conclusion.

Hint: assuming $\displaystyle{a+b\sqrt{2}\neq 0\,,\,\,\frac{m+n\sqrt{2}}{a+b\sqrt{2}}=\frac{ma-2nb+(an-mb)\sqrt{2}}{a^2-2b^2}}$

Tonio

3. We can also use that $Q(D)$ is the smallest field containing $D$ (up to isomorphism) and $\mathbb{Q}(\sqrt{2})$ the smallest field containing $\mathbb{Q}\cup \{\sqrt{2}\}$ (up to isomorphism) .