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I need help solving the following problem.

Let $F$ be a field and $a\in F$.

Show that the quotient ring $F[x]/(x^2-a)$ is isomorphic to the matrix ring $R=\left\{\left(\begin{array}{cc}x&y\\ay&x\end{array}\right)\Bigg|x,y\in F\right\}$.

I understand this might be a very easy problem, but I am still having some issues setting up this isomorphism. I am trying to construct an explicit map from $F[x]/(x^2-a)$ to $R$ and then verify the isomorphism axioms. Should I break this into two cases? One where $a$ is a square in $F$ and one where $a$ is not a square in $F$? Any help would be appreciated.

user1136316
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  • If you want to construct an explicit map, try multiplying two elements of each ring, and compare the forms of the products to guess an explicit map. – Travis Willse Dec 04 '23 at 04:35
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    Is $A=\begin{pmatrix}0&1\a&0\end{pmatrix}$ a square root of $aI=\begin{pmatrix}a&0\0&a\end{pmatrix}$? If so, then the isomorphism would map $b+cx$ to $b+cA=\begin{pmatrix}b&c\ac&b\end{pmatrix}$. – Geoffrey Trang Dec 04 '23 at 05:40
  • See also https://math.stackexchange.com/a/3844191/589 – lhf Dec 04 '23 at 11:40
  • @GeoffreyTrang This is an answer to the question, not just a comment. Please post your answer as an answer. This brings extra visibility to the answer und puts the question off the unanswered list. Also notice that comments are not indexed by the full text search, don't have any revision history, can only be edited for 5 minutes and allow only limited markup. Comments should only be used to clarify, not answer the problem. See What are comments for more information. – Martin Brandenburg Dec 04 '23 at 22:19

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