Exercise 7, § 13

Let E be a $p$-radical extension of a commutative field K of characteristic $p >0$. Suppose that the ring $E\otimes_KE$ is Noetherian.

a) Prove that there exists an integer $k$ such that $E\subset K^{p^{-k}}$ (reason by contradiction: construct a sequence $(x_n)$ in E such that $(x_n\otimes 1-1\otimes x_n)^{p^{n-1}}\not= 0$ and $(x_n\otimes 1-$ $1\otimes x_n$)$^{p^n}= 0)$.

b) Let $F = K(E^p)$, and let $(x_i)_{i\in I}$ be a $p$-basis of E over F (V, §13, No. 1, p. 98). Set $y_i=x_i\otimes 1$ and $z_i=x_i\otimes 1-1\otimes x_i$. Let Λ be the subset of $\mathbf{N}^{(I)}$ consisting of the families $(\alpha_i)_{i\in I}$ such that $\alpha_i< p$ for every $i$. Prove that the elements $\prod_{i\in I}y^{\alpha_i}_iz_i^{\beta_i}$ for $(\alpha_i), (\beta_i)\in \Lambda$ form a basis of the F-vector space $E\otimes_FE$. Deduce that I is finite (observe that the ring $E\otimes_FE$ is Noetherian).

c) Conclude that E has finite degree over K (use Exercise 1, b) of V, §13, p. 170).