Need help in completing the question related to proving these 4 conditions equivalent related to perfect fields
87624
If $\mathrm{char}(K)=p$ is prime, $L/K$ is separable if and only if $K(\alpha) =K(\alpha^p)$ for all $\alpha \in L$
58811
Every algebraic extension of a perfect field is separable and perfect
46565
Show that $\mathbb{Q}(\sqrt{2})$ is a field.
138632
Proof that if $a > 0$, then $\frac{1}{a} > 0$ using just field and order axioms
37723
Proof that $-(-x) = x$ using just the field axioms
127002
Prove that for a nonzero element $a$ of a field $F$ with $q$ elements $a^{-1} = a^{q-2}$
4996
Proving $Gal(K/\mathbb{Q})$ is $D_4$.
7899
Show that if $p$ is prime then $\Bbb Z_p$ is a field
63559
Is there a more complete way of proving this theorem (that I devised)? OR does this theorem already exist and have a name?
145140
Subfield of $\mathbb{Q}(\sqrt[n]{a})$
93328
$\mathbb{F}_p[X]/(X^2+X+1)$ is a field iff $p \equiv 2 \bmod 3$
138436
How to prove reflexivity, symmetry and transitivity for the following relation?
94429
Alternative methods to prove that $Z_m$ is a field under addition and multiplication $\bmod\ m$ iff $m$ is a prime
83009
Showing that $R(x)$ is a proper subset of $R((x))$ if $R$ is a field
10111
Prove or disprove that Q[√2] is a field
18048
Easy criteria to determine isomorphism of fields?
71102
Subfields of $\mathbb{C}$ which are connected with induced topology