If $L\mid K$ is a finite extension of fields then K is perfect iff L is perfect

Solution 1:

Suppose $K$ is not perfect. Then some $a\in K$ has no $p$th root in $K$. I claim that the polynomial $f(x)=x^{p^n}-a$ is irreducible over $K$ for any $n\in\mathbb{N}$. To prove this, let $b$ be a $p^n$th root of $a$ in an extension field of $K$ and note that $f(x)$ factors as $(x-b)^{p^n}$. Let $g$ be the minimal polynomial of $b$ over $K$. Then $g$ is the minimal polynomial of every root of $f$, so $f=g^m$ for some $m$. Since $\deg f=p^n$, $m$ must be a power of $p$; say $m=p^d$. We then have $g(x)=(x-b)^{p^{n-d}}=x^{p^{n-d}}-b^{p^{n-d}}$ and thus $b^{p^{n-d}}\in K$. If $d>0$, we see that $(b^{p^{n-d}})^{p^{d-1}}=b^{p^{n-1}}$ is a $p$th root of $a$ in $K$, which is a contradiction. Thus $d=0$ and $m=1$ so $f=g$ is irreducible.

Now if $L$ is a perfect extension of $K$, then $a$ must have a $p^n$th root in $L$ so $f$ must have a root in $L$. Since $f$ is irreducible, this means $[L:K]\geq \deg f=p^n$. Since $n$ is arbitrary, this means $[L:K]$ must be infinite.

Solution 2:

In the answer of darij grinberg the following equality is given $[L:K]=[L^p:K^p]$ (for the proof notice that $K$-basis of the vector space $L$ correspond to $K^p$-basis of $L^p$ under Frobenius). Using this I think we can answer the problem in a really short way:

As we have the tower of extensions $$K^p\hookrightarrow K\hookrightarrow L=L^p$$ we have that $[L^p:K^p]=[L:K][K:K^p]$ but as $[L:K]=[L^p:K^p]$ we conclude $[K:K^p]=1$.

Solution 3:

Here is an argument that is simpler than @EricWofsey's, although it may well be equivalent to it in some way.

Assume that $L$ is perfect. We must prove that $K$ is perfect.

Assume the contrary. Thus, $K$ (and therefore $L$ as well) has characteristic $p$ for some prime $p$. Consider this $p$. Let $F$ denote the Frobenius endomorphism of any field of characteristic $p$; this sloppy notation is justified because the restriction of a Frobenius endomorphism to a subfield is always the Frobenius endomorphism of that subfield. Note that $F$ is always injective.

We have $F\left(K\right) = K^p \subsetneq K$ (since $K$ is not perfect). Thus, $F^i\left(F\left(K\right)\right) \subsetneq F^i\left(K\right)$ for each $i \geq 0$ (since $F$ is injective, and thus $F^i$ is injective). Hence, for each $i \geq 0$, we have $F^{i+1}\left(K\right) = F^i\left(F\left(K\right)\right) \subsetneq F^i\left(K\right)$. Hence, we have a chain $F^0\left(K\right) \supsetneq F^1\left(K\right) \supsetneq F^2\left(K\right) \supsetneq \cdots$ of fields. Since $L \supseteq K = F^0\left(K\right)$, we can extend this chain to a chain \begin{equation} L \supseteq F^0\left(K\right) \supsetneq F^1\left(K\right) \supsetneq F^2\left(K\right) \supsetneq \cdots . \label{darij1.eq.chain1} \tag{1} \end{equation}

On the other hand, $F\left(L\right) = L^p = L$ (since $L$ is perfect), and thus (by induction) we see that $F^i\left(L\right) = L$ for each $i \geq 0$.

Now, let $d = \left[L : K\right]$. Also, let $i = d+1$. Thus, $d+1 = i < i+1$.

From $d = \left[L : K\right]$, we see that $L$ is a $d$-dimensional $K$-vector space. Hence, $F^i\left(L\right)$ is a $d$-dimensional $F^d\left(K\right)$-vector space (since $F$ is injective, and thus $F^i$ is injective). In other words, $L$ is a $d$-dimensional $F^i\left(K\right)$-vector space (since $F^i\left(L\right) = L$).

Moreover, from \eqref{darij1.eq.chain1}, we obtain \begin{equation} L \supseteq F^0\left(K\right) \supsetneq F^1\left(K\right) \supsetneq F^2\left(K\right) \supsetneq \cdots \supsetneq F^i\left(K\right) . \label{darij1.eq.chain2} \tag{2} \end{equation} This is a chain of fields, and thus is a chain of $F^i\left(K\right)$-vector subspaces of $L$ (because any field is a vector space over any of its subfields). All these subspaces have dimension $\leq d$ (since $L$ is a $d$-dimensional $F^i\left(K\right)$-vector space), and these dimensions must decrease by at least $1$ at each $\supsetneq$ sign. Thus, the chain \eqref{darij1.eq.chain2} yields the following chain of inequalities: \begin{equation} d \geq \dim\left(F^0\left(K\right)\right) > \dim\left(F^1\left(K\right)\right) > \dim\left(F^2\left(K\right)\right) > \cdots > \dim\left(F^i\left(K\right)\right) \end{equation} (where $\dim$ stands for dimension as $F^i\left(K\right)$-vector spaces). Thus, the $i+1$ integers $\dim\left(F^0\left(K\right)\right),\dim\left(F^1\left(K\right)\right),\dim\left(F^2\left(K\right)\right),\ldots,\dim\left(F^i\left(K\right)\right)$ must be distinct and must all belong to the set $\left\{0,1,\ldots,d\right\}$. Thus, this set $\left\{0,1,\ldots,d\right\}$ must contain at least $i+1$ distinct integers. But it does not (since its size is $d+1 < i+1$). This contradiction shows that our assumption was wrong. Hence, we have shown that $K$ is perfect.