I was reading wikipedia for Galois groups and this term suddenly appears and there is no definition for it.

What is a Galois closure of a field $F$? Does this mean a maximal Galois extension of $F$ so that it merely means a separable closure of $F$?

Secondly, what is a Galois group of an arbitrary extension $E/F$?

Wikipedia states that $Gal(E/F)$ is defined as $Aut(G/F)$ where $G$ is a Galois closure of $E$.

(Since I don't know what Galois closure is, if you don't get bothered, i will add this part after I know what a Galois closure is. Otherwise, I will post another one)


Two points: One, Galois closure is a relative concept, that is not defined for a filed, but for a given extension of foields.

Second, it is not something maximal. To the contrary it is something minimal.

Given an extension of fields $F\subset E$ if it is not Galois, then the smallest extension of $F$ that containing $E$ and that is a Galois extn of $F$ is called the Galois closure.

  • $\begingroup$ How do I define "smallest"? For example, "maximal" algebraic extension means that for any algebraic extension $E$ of $\bar F$, $E=\bar F$. How do I define smallest in this manner? $\endgroup$ – Rubertos May 24 '15 at 10:42
  • 2
    $\begingroup$ If many extension fileds over a given field are all Galois extensions, then their intersections is again a field, contains the base filed and is a Galois $\endgroup$ – P Vanchinathan May 24 '15 at 11:30

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