0

Let $k$ be an imperfect field of characteristic $p$. Let $a \in k$ be an element that is not a $p$-th power. How to prove the ring $$R:=k[x,y]/(x^p+y^p-a)$$ is a regular ring?

(On the other hand, if $a$ is a $p$-power, then $(x^p+y^p-a)$ is a $p$th- power in $k[x,y]$ so the ring $R$ is not regular as it is not even reduced.)

The Jacobian of $x^p+y^p-a$ is $[0\ 0]$ and is independent of $a$, so I think Jacobian criterion doesn't distinguish the two cases.

That $R$ is regular is basically stated at https://en.wikipedia.org/wiki/Geometrically_regular_ring

usr0192
  • 3,157
  • 2
    Putting $u=x+y$, $R=k[u,y]/(u^p-a)$ and $k[u]/(u^p-a)$ is a field. – Mohan Sep 07 '20 at 16:16
  • @Mohan Nice! In case anyone encounters this question later, I had to look up why $k[u]/(u^p-a)$ is a field...that's true for k of any characteristic (and $a$ not a $p$th power in $k$) https://math.stackexchange.com/questions/403924/xp-c-has-no-root-in-a-field-f-if-and-only-if-xp-c-is-irreducible – usr0192 Sep 07 '20 at 18:04

0 Answers0