What is a Galois closure and Galois group? 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)
 A: Two points: One, Galois closure is a relative concept, that is not defined for a field, but for a given extension of fields.
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.
EDIT (added on 8th Sep 2020)
Given a finite extension $E$ of $F$ we can obtain that as  $F[X]/(f(X))$ for an irreducible polynomial $f(X)$ with coefficients in the field $F$. So $E$ is one field that contains a root of $f(X)$. Now the Galois closure is  theoretically   the field generated  by all the roots of $f(X)$.
Example: Let $b=\sqrt[3]{2}$ the positive real cube root of 2. So the field $E=\mathbf{Q}[b]$ is an extension of degree 3 over $F=\mathbf{Q}$ completely contained inside the real numbers. The $f(X)$ in this case is $X^3-2$. This cubic polynomial has 2 other roots in the complex number system. Let $\omega= \cos 120^\circ + i\sin 120^\circ$ be a cube root of 1. Then $\omega b, \omega ^2b$ are the other two roots of $f(X)$. The new field $K=\mathbf{Q}[b,\omega]$ is the Galois closure of $E$ over $\mathbf{Q}$. (also called the splitting field of $f(X)$ over the rationals. So normal closure for $E=F[X]/(f(X))$ can be defined as the splitting field for the polynomial $f(X)$.
