The Stacks project

Example 33.6.10. Let $k = \mathbf{F}_ p(s, t)$, i.e., a purely transcendental extension of the prime field. Consider the variety $X = \mathop{\mathrm{Spec}}(k[x, y]/(1 + sx^ p + ty^ p))$. Let $k'/k$ be any extension such that both $s$ and $t$ have a $p$th root in $k'$. Then the base change $X_{k'}$ is not reduced. Namely, the ring $k'[x, y]/(1 + s x^ p + ty^ p)$ contains the element $1 + s^{1/p}x + t^{1/p}y$ whose $p$th power is zero but which is not zero (since the ideal $(1 + sx^ p + ty^ p)$ certainly does not contain any nonzero element of degree $< p$).

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