Show that $[K:F]=p^2$ and the separability and inseparability degrees of $K/F$ are both equal to p

Question

Let p be prime, $F_p$ be the prime field of p elements, $X$ and $Y$ algebraically independent variables over $F_p$, $K=F_p(X,Y)$, and $F=F_p(X^p-X,Y^p-X)$.

(a) Show that $[K:F]=p^2$ and the separability and inseparability degrees of $K/F$ are both equal to $p$

(b) Show that there exists a field $E$, such that $F\subset E\subset K,$ which is a purely inseparable extension of $F$ of degree $p$.

Can someone tell me how to prove this question? I am a little confused how to analyze this question!

Answer 1

Extended hints:

• Show that $X$ is a zero of $m(T)=T^p-T-(X^p-X)\in K[T]$.
• Show that the zeros of $m(T)$ are $X+i,i=0,1,\ldots,p-1$.
• Show that $m(T)$ is irreducible in $K[T]$. If related facts have not been explained in class then you have to work hard at this point. Or search the site with the buzzword Artin-Schreier.
• Show that $F(X)/F$ is Galois and degree $p$.
• Show that $(X-Y)^p\in F$.
• Find $E$.