If $F^p = F$ and $E/F$ is algebraic, then $E/F$ is separable and $E^p = E$ : Corollary V.6.12 from Lang's *Algebra*
58811
Every algebraic extension of a perfect field is separable and perfect
19202
$\mathbb{F}_{p^d}\subseteq\mathbb{F}_{p^n}$ if and only if $d$ divides $n$
26677
thoughts about $f(f(x))=e^x$
84206
How do I show that if $f$ is entire and $\{\lvert f(z)\rvert < M\}$ is connected for all $M$, then $f$ is a power function?
146591
Inseparable polynomial over a non-perfect field
136943
A problem with a proof about formal series in Lang's Algebra
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If $\alpha$ is an algebraic element and $L$ a field, does the polynomial ring $L[\alpha]$ is also a field?
85096
Let $K$ be a field and $f(x)\in K[x]$. Prove that $K[x]/(f(x))$ is a field if and only if $f(x)$ is irreducible in $K[x]$.
89782
To show that $\langle x-a , y-b\rangle$ is a maximal ideal of $F[x,y]$ by showing that $F[x,y]/\langle x-a , y-b\rangle$ is a field
106091
Showing $\mathcal{P}(X)$ is isomorphic to $2^X$
139874
If $X^\ast $ is separable $\Longrightarrow$ $S_{X^\ast}$ is also separable
134859
Let $(X, \mathcal{T})$ be a metrizable space. Prove that $X$ Lindelöf implies that $X$ is separable.
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Show that $[K:F]_s = [K:L]_s [L:F]_s$ and $[K:F]_i = [K:L]_i [L:F]_i$.
73419
For $f$ entire and $\{f(kz) : k \in \mathbb{C}\}$ a normal family, prove that $f$ is a polynomial.
49438
Prove that $X^\ast$ separable implies $X$ separable
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$f(x)=1/q$ for $x=p/q$ is integrable
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Every finite extension of a finite field is separable