# The proof of the separability of a dominant morphism of irreducible varieties

Let $T$ be a torus, $m$ a positive integer. Prove that the morphism $x \mapsto x^m$ maps onto $T$, and is separable if $(p,m)=1$.

Here, $p$ $(>0)$ is the characteristic of the base field. A dominant morphism $\phi: X \rightarrow Y$ of irreducible algebraic varieties is called separable if the field extension $K[X]/\phi^*(K[Y])$ is separable.

It seems not easy to prove the separability of the field extension.

Many thanks.

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Dear ShinyaSakai, In fact it is pretty easy to prove if you know all the relevant information. As a warm-up, can you prove it in the case when $T$ is a $1$-dimensional split torus? Also, what is your definition of a torus in general? Regards, –  Matt E Jul 27 '11 at 15:22
Thank you very much. I am very sorry for being unclear. In the book I am reading, a torus is defined to be a closed connected diagonalizable subgroup of an algebraic group. –  ShinyaSakai Jul 27 '11 at 16:44
Dear ShinyaSakai, Okay. Now, can you prove this separability statement in the case when $T = \mathbb G_m$? Regards, –  Matt E Jul 27 '11 at 17:01
Dear Matt E, thanks a lot. $K[\mathbb{G}_m]=K[x,x^{-1}]$, and $\phi^*K[\mathbb{G}_m]=K[x^m,x^{-m}]$. The extension $K[x,x^{-1}]/K[x^m,x^{-m}]$ is separable in case $(p,m)=1$. The next step is extending to the general case, isn't it? –  ShinyaSakai Jul 29 '11 at 3:50
Dear @Matt E: thanks a lot. $K[\mathbb{G}_m] = K[x,x^{-1}]$, and $\phi^*K [\mathbb{G}_m] = K [x^m, x^{-m}]$. The extension $K[x,x^{-1}]/ K[x^m,x^{-m}]$ is separable in the case $(p,m) =1$. The next step is extending to the general case, isn't it? –  ShinyaSakai Oct 17 '11 at 14:19