Let $k$ be a field , $R=k[X] , f \in R$ . Let $A=k[X,Y]/(Y^2-f)$ . I know $A$ is an integral domain iff $f$ is not a perfect square and hence forth assume $A$ is a domain. If $f$ has a square factor, then I can show that $A$ is not a normal domain. My question is : If $f$ has no square factor, then how to show that $A$ is a normal domain ?
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All you need can be found in this answer. – user26857 Nov 22 '17 at 07:23
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Btw, the result holds for characteristic of $k$ different from $2$. Otherwise this can be wrong. – user26857 Nov 22 '17 at 07:27