$L/K$ be a field extension with $Char(K) = p > 0$ and $[L : K] = n$ cannot be divided by $p$. Show that $L/K$ is separable.

Question

Hey I want to check my solutions for this exercise:

Let $K$ be a field with $Char(K) = p > 0$ and let $L/K$ be an extension whose degree $[L : K] = n$ cannot be divided by $p$. Show that $L/K$ is separable.

So my idea was: Let $f$ be a irreducible polynomial in $L$

We have for $f(x):=x^n+a_{n-1}x^{n-1}+\dots+a_{0}$

$f'(x)=nx^{n-1}+(n-1)a_{n-1}x^{n-2}+\dots+a_{1}$

Since $p$ cannot divide $n$, $f'\neq 0$ and therefore the field extension is separable.

I think that I have made some mistakes.

Can someone check my solution?

Answer 1

You're close. One mistake is that you supposed that the degree of $f$ is equal to $n$, which may not be. What is for sure, is that the degree $m$ of $f$ divides $n$. Without loss of generality, we can suppose $m > 1$.

Then you can apply your argument as if $p$ would divide $m$, it would also divide $n$.

Answer 2

I do not see why you select an arbitrary irreducible polynomial $f$ in $K[x]$. (Though you wrote "Let $f$ be a irreducible polynomial in $L$", I guess you meant $f\in K[x]$.)

I guess you actually meant:

Proof: Select an arbitrary element $a\in L$. Let $f(x)$ be the (monic) irreducible polynomial of $a$ over $K$. Assume \begin{equation*} f(x)=x^m+a_{m-1}x^{m-1}+\cdots+a_0. \end{equation*} Sincce $K\subseteq K(a)\subseteq L$, we have $m=[K(a):K]$ is a divisor of $n$, and hence $p$ is not a divisor of $m$. Therefore $f'(x)=mx^{m-1}+\cdots\ne 0 $$\Rightarrow$ $a$ is seperable over $K$. Since $a$ is an arbitrary element in $L$, we infer $L/K$ is separable.

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By the way, if you know the proposition of purely inseparable extension, then here is another approach: Let $F$ be the collection of elememnts in $L$ separable over $K$: \begin{equation*} F:=\{ x\in L|\ x \text{ is separable over $K$} \}. \end{equation*} Then $F$ is a field, and we have successive field extension \begin{equation*} K\subseteq F\subseteq L, \end{equation*} where $F/K$ is a separable extension and $L/F$ is a purely inseparable extension, with $[L:F]=p^k$ for some $k$. Since $n=[L:K]=[L:F][F:K]=p^k[F:K]$, we must have $k=0$, i.e. $L=F$ $\Rightarrow$$L/K$ is separable.