# 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)

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.
• 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? – Rubertos May 24 '15 at 10:42