# Need help in completing the question related to proving these 4 conditions equivalent related to perfect fields

Consider the following question:

(a) The following conditions on a field K are equivalent :So, Prove them

(i) every irreducible polynomial in K[x] is separable ;

(ii) every algebraic closure K of K is Galois over K;

(iii) every algebraic extension field of K is separable over K;

(iv) either char K = 0 or char K = p and $$K = K^p$$.

(i)=>(ii) This part I am not able to do. What i am missing is how to prove that for all $$\sigma \in Aut_K{\bar K}$$ only y$$\in K$$ gets fixed and rest all are not fixed.

(ii)=> (iii) I have done as if an algebraic extension is galois over K and then the extension is also separable over K ( This is a theorem).

(iii)=>(iv) I tried by assuming that char K $$\neq 0$$, so, As field is integral domain, so Characterstic must be prime and so, let char K=p but then I am not able to move towards proving $$K=K^{p}$$.

(iv)=>(i) I have solved . So, I don't require help with that.

Kindly shed some light on this

• What are you allowed to use? That $f$ is separable iff $gcd(f,f')=1$? Jan 22, 2021 at 14:43
For (i)=>(ii): Let $$L$$ be an algebraic closure of $$K$$. Then $$L/K$$ is by definition normal and by assumption separable. Depending on your definition of Galois-extension we might be done here. If your definition of Galois extension is $$K=L^{Aut_KL}$$, then look at the answer to one of your previous questions.
For (iii)=>(iv): If $$a\in K$$ there is some $$b\in L$$ in an extension field with $$b^p=a$$. Now look at the irreducible factors (over $$K$$) of the polynomial $$x^p-a=(x-b)^p$$ and use your assumption.
• Why is in $(iii) \to (iv)$ there exists some $b\in L$ in an extension field with $b^p =a$? can you please tell? Oct 17, 2021 at 6:17