Profinite group galois group pdf

This is the content of three lectures at the winter school on galois. A profinite group is a topological group that is isomorphic to the inverse limit of an inverse system of discrete finite groups. But they have an interesting theory in their own right. This construction has a generalization in algebraic geometry in the definition of the fundamental group of a scheme.

Much of recent research on abstract infinite groups is related to profinite groups because residually finite groups are naturally embedded in a profinite group. Now we know they include valuable galois extensions of the rationals that present its absolute galois group through known groups. Pro nite groupsarise in nature as galois groups of in nite algebraic extensions. The system forms in a neighbourhood base of the identity cf. Profinite groups a directed set iis a partially ordered set such that for all i. Profinite groups and infinite galois extensions jonatan lindell. In general, a profinite group which is generated by one element is procyclic.

Artins theorem on finite automorphism groups of fields extends to profinite groups, and hence every profinite group is a galois group. A large class of actions of profinite groups on cantor sets is given by arboreal representations of absolute galois groups of. A profinite group is an inverse limit of a system of finite groups the finite groups are considered as compact discrete topological groups and so the inverse limit, as a closed subspace of the compact space that is the product of all those finite groups has the inverse limit topology, hence is, as is said above, a compact hausdorff, totally disconnected group. The corresponding profinite group is isomorphic to, thus can be considered as a profinite group. In particular, we study the procyclic groups zp and z and prove that every finite field has. Z p, where p ranges over all primes and each z p is either zp n z for some n. Profinite groups can be defined in either of two equivalent ways. Properties of the topological group re ect grouptheoretic properties of all the nite groups. An enveloping ellis group of such an action is a profinite group.

We treat g k as a fundamental object of study because it allows us to control all separable extensions l of k in one stroke. However, the most prominent example for a profinite group is the galois group of a galois extension nk. Galois theory o ers a natural frame in order to describe galois groups as pro nite groups. A pro nite group is a compact topological group that is built out of nite groups. In particu lar, the profinite freeness of a closed subgroup provides us with a solution of the. A pro nite space group is the projective limit of nite sets groups. Each profinite group actually appears as a galois group. Profinite groups and infinite galois extensions diva portal. Pdf profinite groups are galois groups semantic scholar. This gave a correspondence between closed subgroups and intermediate elds. Consequently a big part of this project consists in studying topological and profinite groups.

Now we know they include valuable galois extensions of the rationals that present its absolute galois group through known. In this context, an inverse system consists of a directed set. Free subgroups of finitely generated free profinite groups. In the 1930s wolfgang krull extended the fundamental theorem of galois theory to in nite galois extension via introducing a topology on the galois group. As such they are of interest in algebraic number theory. It is well known that every finite group is the galois group of some field extension, but the corresponding statement about profinite groups. An inverse system of groups is a collection of groups fg igindexed by a directed set itogether with group homomorphisms.