Galois extension

2020 Mathematics Subject Classification: Primary: 12F10 [MSN][ZBL]

of a field

An extension of a field that is algebraic, normal and separable. The group of all automorphisms of a Galois extension $K / k$ that leave all elements of $k$ invariant is called the Galois group of this extension, denoted by $\def\Gal{\textrm{Gal}}\Gal(K/k)$. The study of these groups is a major part of Galois theory.

An alternative characterisation of Galois extensions is that an extension $K/k$ is Galois if, taking $G$ to be the group of automorphisms of $K$ that leave all elements of $k$ fixed, then the subfield of $K$ fixed by $G$ is exactly $k$.

In the case of finite extensions, the order (number of elements) of $G = \Gal(K/k)$ is equal to the degree of $K$ over $k$. To each subgroup $H$of $G$ corresponds a subfield $P = K^H$ of $K$, consisting of all elements from $K$ that remain fixed under all automorphisms from $H$. Conversely, to each subfield $P\subset K$ that contains $k$ corresponds a subgroup $H = G_P$ of $G$, consisting of all automorphisms leaving each element of $P$ invariant. Here, $K$ is a Galois extension of $P$ and $\Gal(K/P)=H$. The main theorem in Galois theory states that these correspondences are mutually inverse, and are therefore one-to-one correspondences between all subgroups of $\Gal(K/k)$ and all subfields of $K$ containing $k$. In this correspondence certain properties of subgroups correspond to the properties of subfields and vice versa. Thus, a subgroup $H$ will be a normal subgroup of $\Gal(K/k)=G$ if and only if the field $P$ which corresponds to it is a Galois extension of $k$. Moreover, $\Gal(P/k)$ is isomorphic to $G/H$.

For infinite extensions, define the Krull topology on the group $G$ by taking a basis of the open neighbourhoods of the identity to be the normal subgroups of finite index. There is then a one-to-one correspondence between the closed subgroups of $G$ and the subfields of $K / k$. Open subgroups of $G$ correspond to subfields of $K$ that have finite degree over $k$. If $H$ is an arbitrary subgroup of $G$, then the extension $K/K^H$ is Galois and has the closure of $H$ as Galois group. Cf. Galois topological group.

References

• Kaplansky, Irving Fields and rings (2nd ed.) University of Chicago Press (1972) ISBN 0-226-4241-0 Zbl 1001.16500
• Lang, Serge Algebra (3rd rev. ed.) Graduate Texts in Mathematics 211 Springer (2002) Zbl 0984.00001