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.
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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 $\displaystyle \displaystyle{a+b\sqrt{2}\neq 0\,,\,\,\frac{m+n\sqrt{2}}{a+b\sqrt{2}}=\frac{ma-2nb+(an-mb)\sqrt{2}}{a^2-2b^2}}$ Tonio
We can also use that $\displaystyle Q(D)$ is the smallest field containing $\displaystyle D$ (up to isomorphism) and $\displaystyle \mathbb{Q}(\sqrt{2})$ the smallest field containing $\displaystyle \mathbb{Q}\cup \{\sqrt{2}\}$ (up to isomorphism) .
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