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