1

There is another post with this question which ask for a proof not using embeddings: If $\alpha$ separable over $F$ then $F(\alpha )/F$ is a separable extension..

I would like just to know the traditional proof using embeddings, as my lecture notes in my fields and Galois theory course left this reasoning for the reader.

My reasoning is the following:

With embeddings we know that $K(\alpha)/K$ is separable if and only if $|Hom(K(\alpha)/K,\bar{K}/K)|=[K(\alpha):K]$, and this is guaranteed cause $Irr(\alpha,K)$ has $n=[K(\alpha):K]$ distinct roots thanks to separability, let them be $\{ \alpha_i \}_{i=1}^{n}$ and for each we can construct a unique embedding such that $\alpha\longmapsto \alpha_i$, so there is exactly $[K(\alpha):K]$ of these embeddings.

Is there any nuance I should also consider?

IAG
  • 255
  • Do you know how to prove such embeddings exist? This is not very trivial, and you just stated it as a fact. – Mark Apr 11 '24 at 15:53
  • @Mark I know that it exists thanks to a result stated in previous sections of my course in which it was proved quite easely this way: $K(\alpha)\cong \dfrac{K[X]}{f(X)} \cong K(\alpha_i)$ where $f(X)=Irr(\alpha,K)=Irr(\alpha_i,K)$. – IAG Apr 11 '24 at 16:11
  • An then you just extend that isomorphism to the algebraic closure $\bar{K}$ so you have an embedding such that $\alpha \mapsto \alpha_i$. – IAG Apr 11 '24 at 16:13
  • If you know that, then this is correct. I guess you don't even need to extend to $\overline{K}$. You need embeddings $K(\alpha)\to\overline{K}$. – Mark Apr 11 '24 at 16:25
  • @Mark I mean, extending in the codomain, just composing with the inclusion. – IAG Apr 11 '24 at 16:33
  • By “imbedding” I guess you mean following: Let $E$ and $F$ each be extension fields of $K$ and Let $u\in E$ and $v\in F$ be algebraic over $K$. Then $u$ and $v$ are roots of the same irreducible polynomial $f\in K[x]$ if and only if there is an isomorphism of fields $K(u)\cong K(v)$ which sends $u$ onto $v$ and is the identity on $K$”. – user264745 Jun 22 '24 at 18:51
  • I give a summary of all the approaches I know of for this problem here here https://math.stackexchange.com/a/5050132/405572 – D.R. Mar 27 '25 at 03:45

0 Answers0