I'm a little bit confused by the statement of this corollary, (Corollary 23 pg 594 of Dummit and Foote)

Let $E/F$ be any finite separable extension. Then $E$ is contained in an extension $K$ which is Galois over $F$ and is minimal in the sense that in a fixed algebraic closure of $K$ any other Galois extension of $F$ containing $E$ contains $K$.

What does he mean exactly by saying "in a fixed algebraic closure of $K$..."?

  • $\begingroup$ The algebraic closure is only unique up to isomorphism. If $E$ is an extension of $F$, then the algebraic closure of $E$ is "the same" as the algebraic closure of $F$ since $F$ is embedded in $E$. Does that answer your question? $\endgroup$ – user194928 Jun 14 '15 at 22:20
  • $\begingroup$ I get it now, thanks. $\endgroup$ – TuoTuo Jun 14 '15 at 22:41

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