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1/ If $d$ is not a square in $\mathbb{Q}$, show that $\mathbb{Q}[\sqrt{d}]\approxeq\mathbb{Q}[X]/$ where $(X^2 - d)$ is the principal ideal of $\mathbb{Q}[X]$ generated by $X^2 - d$.

2/ If $d_1, d_2$, and $d_1/d_2$ are not squares in $\mathbb{Q}\backslash\left\{ 0\right\} $, show that $\mathbb{Q}[\sqrt{d_1}]$ and $\mathbb{Q}[\sqrt{d_2}]$ are not isomorphic.

3/ Let $R_1 = \mathbb{Z}_5[X]/$ and $R_2 = \mathbb{Z}_5[X]/(X^2 - 3)$, then the statement $R_1\approxeq R_2$ true or false?.

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    Is this homework? (The phrasing makes it sound like it is). What have you tried?2012-10-26
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    $\varphi_d:\mathbb{Q}[X]\to\mathbb{Q}[\sqrt{d}]$ give by $\varphi_d(a_0+a_1X+\ldots+a_nX^n)=a_0+a_1\sqrt{d}+\ldots+a_n(\sqrt{d})^n$, then $\varphi_d$ is homomorphism, and $\varphi_d(X^2-d)=0$ then $ \subset Ker \varphi_d$, but why $ = Ker \varphi_d$2012-10-26
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    nobody help me?2012-10-26

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