Galois group

2020 Mathematics Subject Classification: Primary: 12F10 Secondary: 11R32 [MSN][ZBL]

The automorphism group of a Galois extension $L$ of a field $k$, i.e. the group of all automorphisms of the field $L$ leaving the elements of the subfield $k$ fixed. The group is denoted by $G(L/k)$ or by $\textrm{Gal}(L/k)$. The field of invariants $L^G(L/k)$ coincides with the field $k$. If $L$ is the splitting field of a polynomial $f$ over $k$, the Galois group $G(L/k)$ is also called the Galois group of the polynomial $f$. These groups are important in the Galois theory of algebraic equations. The computation of the Galois groups for extensions of algebraic number fields is one of the fundamental tasks of algebraic number theory. Finding Galois extensions with an Abelian Galois group (Abelian extensions) is a part of class field theory. Galois groups of algebraic function fields are also a subject of algebraic geometry.

Let $L$ be a field and let $G$ be a finite subgroup of the automorphism group of $L$; $L$ will then be a Galois extension of the field of invariants $k=L^G$, and the Galois group of this extension is isomorphic to $G$; moreover, the degree of the extension, $[L:k]$, is equal to the order of $G$.

The fundamental result on Galois groups is the following theorem, which is sometimes called the main theorem on Galois extensions or the theorem on Galois correspondence. If $L$ is a Galois extension of finite degree of a field $k$, then there exists a one-to-one correspondence between all subgroups $H$ of the Galois group $G(L/k)$ and all subfields $F$ of $L$ that contain $k$, and the $H$ and $F$ corresponding to each other are such that $F$ is the field of invariants of $H$ and $H$ is the Galois group of $G(L/k)$ (cf. Galois correspondence). This theorem has numerous analogues in many mathematical theories, and can be generalized to extensions of infinite degree (cf. Galois topological group). There exists a generalization of the concept of a Galois group to extensions of arbitrary commutative rings, schemes (cf. Fundamental group), and also to the case of extensions of skew-fields.

References

[Bo]	N. Bourbaki, "Algebra", Elements of mathematics, 1, Springer (1989) pp. Chapt. 1–3 (Translated from French) MR1727844 Zbl 1111.00001
[Ja]	N. Jacobson, "The theory of rings", Amer. Math. Soc. (1943) MR0008601 Zbl 0060.07302
[La]	S. Lang, "Algebra", Addison-Wesley (1984) MR0783636 Zbl 0712.00001
[Po]	M.M. Postnikov, "Fundamentals of Galois theory", Noordhoff (1962) (Translated from Russian) MR0136603 Zbl 1092.12005

Comments

A Galois group can be endowed with the Krull topology, making it a topological group. This topology is discrete if and only if the group is finite. For the Galois correspondence in the case of infinite Galois groups see Galois topological group.