nLab Galois group

see at Galois theory for more

Idea

Given a field extension one can consider the corresponding automorphism group. The main statement of Galois theory is that, when the field extension is Galois, this group is called the Galois group and its subgroups correspond to subextensions of the field extension.

In algebraic geometry, Grothendieck defined an analogue of the Galois group called the etale fundamental group of a connected scheme.

Even more generally there is an analogue of the Galois group in stable homotopy theory. In fact one can define the Galois group of any presentable symmetric monoidal stable (infinity,1)-category, and there is an analogue of the Galois correspondence. In particular one gets a Galois group associated to an E-infinity ring spectrum. One recovers the Galois group of a scheme as the Galois group of its derived category of quasi-coherent sheaves.

Definition

Definition

Let $K\hookrightarrow L$ denote a Galois field extension, then the automorphism group

$$Gal(K\hookrightarrow L):=Aut_K(L)$$

consisting just of those automorphisms of $L$ whose restriction to $K$ is the identity is called Galois group of the field extension $K\hookrightarrow L$.

Every Galois group $Gal(K\hookrightarrow L)=lim_{K\hookrightarrow E\hookrightarrow L}Gal (K\hookrightarrow E)$ is a profinite topological group in that it is the limit of the topologically discrete Galois groups of the intermediate finite extensions between $K$ and $L$.

The just defined Galois group is the one occurring in the classical Galois theory for fields. The analog of the Galois group in Galois theory for schemes is a fundamental group (of a scheme) and is rarely called a ‘’Galois group’’.

The Galois group $Gal(K\hookrightarrow K_s)$ of the separable closure of $K$ is called the absolute Galois group of $K$. In this case we have $Gal(K\hookrightarrow K_s)\simeq \pi_1(Spec\; K)$ is equivalent to the fundamental group of the scheme $Spec K$. In particular the notion of fundamental group (of a point of) a topos generalizes that of Galois group. This observation is the starting point and motivating example of Grothendieck's Galois theory and more generally of that of homotopy groups in an (infinity,1)-topos.

If the scheme, moreover, is a group scheme (i.e. endowed with a group structure) modules over the Galois group, which are called Galois modules, play an important role in algebraic number theory.

Properties

Relation to étale fundamental group

For $K$ a field, then the absolute Galois group of $K$ is equivalent to the étale fundamental group/algebraic fundamental group of the spectrum of $K$.

$$\pi_1(Spec(K)) \simeq Gal(K_{sep}/K) \,.$$

If $K$ is a number field, write $\mathcal{O}_K$ for its ring of integers, so that $Spec(\mathcal{O}_K)$ is an arithmetic curve. Then

$$\pi_1(Spec(\mathcal{O}_K)) \simeq Gal(K_{alg}^{ur}/K) \,,$$

where $K_{alg}^{ur}$ is the maximal algebraic extension of $K$ that is unramified at all non-zero prime ideals (e.g. Lenstra 85, Example 1.12).

See also at Galois theory – Statement of the main theorem.

Relation to Frobenius maps

For $K$ a number field then the Frobenius maps induce canonical elements in the Galois group. See at Frobenius morphism – As elements of the Galois group.

This crucially enters the definition of Artin L-functions associated with Galois representations.

Related concepts

• Galois theory

• motivic Galois group

References

Monographs

• Julio R. Bastida, Field Extensions and Galois Theory, Cambridge University Press (1984) [doi:10.1017/CBO9781107340749]

• Hendrik Lenstra, Galois theory for schemes , Mathematisch Instituut Universiteit van Amsterdam (1985) [pdf]

For the Galois group in stable homotopy theory, see

• Akhil Mathew, The Galois group of a stable homotopy theory, arXiv.

category: Galois theory