# Krull topology

The Krull topology , according to Wolfgang Krull , is a topology on the Galois group of a not necessarily finite field extension , so that this becomes a so-called topological group . ${\ displaystyle L / K}$ ## Definition of Galois extensions

Let it be a not necessarily finite Galois extension of the body. For an infinite extension, Galois means that the extension is separable and for every finite Galois partial extension also contains the normal envelope of . ${\ displaystyle L / K}$ ${\ displaystyle K \ subseteq M \ subseteq L}$ ${\ displaystyle M}$ There are several ways to define the Krull topology:

1. The ambient basis of the neutral element is defined as the set

${\ displaystyle \ {G (L / M) \ mid K \ subseteq M \ subseteq L, \ [M: K] <\ infty \}}$ of the Galois groups for over finite partial extensions . ${\ displaystyle K}$ ${\ displaystyle M}$ 2. There is a canonical bijection

${\ displaystyle G (L / K) = \ lim _ {M} G (M / K),}$ where all passes through finite partial extensions . If one provides the finite groups with the discrete topology and the projective limits with the limit topology, then one obtains the same topology as under 1. With this representation it can be seen that there is a pro- finite group . ${\ displaystyle M}$ ${\ displaystyle K}$ ${\ displaystyle K \ subseteq M \ subseteq L}$ ${\ displaystyle G (M / K)}$ ${\ displaystyle G (L / K)}$ ## Law of Galois Theory

The importance of the Krull topology lies in the fact that it enables the main theorem of Galois theory to be extended to infinite Galois extensions: If there is an infinite Galois extension, then there is a canonical bijection between partial extensions and closed subgroups of : The subgroup corresponds to an extension${\ displaystyle L / K}$ ${\ displaystyle K \ subseteq M \ subseteq L}$ ${\ displaystyle G (L / K)}$ ${\ displaystyle M}$ ${\ displaystyle G (L / M) \ subseteq G (L / K),}$ a subgroup the extension ${\ displaystyle U \ subseteq G (L / K)}$ ${\ displaystyle L ^ {U} = \ {x \ in L \ mid \ sigma x = x \ \ mathrm {f {\ ddot {u}} r \ alle} \ \ sigma \ in U \}.}$ A partial expansion is normal (and thus Galois) if and only if a normal sub-divisor is in; the Galois group is canonically isomorphic to the quotient . ${\ displaystyle M / K}$ ${\ displaystyle G (L / M)}$ ${\ displaystyle G (L / K)}$ ${\ displaystyle G (M / K)}$ ${\ displaystyle G (L / K) / G (L / M)}$ ## Representations

Let it be a body and a separable conclusion of . Further let us be a vector space (over any field). If one provides the discrete topology, then representations of on are continuous if and only if they factorize over a finite quotient for a finite extension . The category of the continuous representations of is in this sense the union of all categories of representations of the groups for finite extensions . ${\ displaystyle K}$ ${\ displaystyle K ^ {\ mathrm {sep}}}$ ${\ displaystyle K}$ ${\ displaystyle V}$ ${\ displaystyle \ mathrm {GL} (V)}$ ${\ displaystyle G (K ^ {\ mathrm {sep}} / K)}$ ${\ displaystyle V}$ ${\ displaystyle G (M / K)}$ ${\ displaystyle M / K}$ ${\ displaystyle G (K ^ {\ mathrm {sep}} / K)}$ ${\ displaystyle G (M / K)}$ ${\ displaystyle M / K}$ ## Generalization: Not algebraic extensions

Let it be any extension of the body. The Krull topology on the group of body automorphisms of which fix element by element is the topology for which the subgroups ${\ displaystyle L / K}$ ${\ displaystyle \ operatorname {Aut} (L / K)}$ ${\ displaystyle L}$ ${\ displaystyle K}$ ${\ displaystyle G (S) = \ {\ sigma \ in \ operatorname {Aut} (L / K) \ mid \ sigma s = s \ \ mathrm {f {\ ddot {u}} r \ all} \ s \ in S \}}$ for finite subsets form an environment basis of the element. becomes a topological group with this topology. ${\ displaystyle S \ subseteq L}$ ${\ displaystyle \ operatorname {Aut} (L / K)}$ ## Individual evidence

1. ^ Ina Kersten : Brewer groups . Universitätsdrucke Göttingen, 2007, ISBN 978-3-938616-89-5 , §15.2 ( online [accessed on January 26, 2017]).