Helmut Koch

Reelle Funktionen einer Veränderlichen

Wir haben in Abschnitt 5.11 den Begriff der reellen Funktion einer Veranderlichen als Abbildung eines abgeschlossenen Intervalls

Group Algebras of pro-p Groups

\left[ {a_1 ;a_2 } \right]: = \left\{ {a \in \mathbb{R}|a_1 \leqslant a \leqslant a_2 } \right\}

Cohomology of Profinite Groups

der Zahlengeraden in die Menge der reellen Zahlen eingefuhrt.

Presentation of pro-p Groups

We now consider as our base fields global fields k of finite type, i.e., finite extensions k/ℚ or k/F q (x). The results of this chapter form the central part of this book. We are going to study the Galois group Gs of the maximal p-extension of k unramified outside some set of primes S of k, and we investigate in particular how to use the theorems on the structure of the Galois group of the maximal p-extension of a local field derived in the last chapter to get results on Gs.

On p -Class Groups and p -Class Field Towers

The results to be proved in this chapter refer again mostly to presentations of a pro-p group by generators and relations. They will be derived using the complete group algebra of a pro-p group. In this chapter, Λ will always denote a compact commutative ring with identity. For finite groups G, the additive group of the group ring Λ[G] is isomorphic to the direct sum of #G copies of Λ, and thus inherits a topology that makes Λ[G] into a compact group ring.

Die komplexen Zahlen

The maximal p-extension \(\hat k\) of a field k is the compositum (inside a fixed algebraic closure of k) of all finite p-extensions of k, i.e., of all normal (separable) extensions of k with p-power degree.

Results from Algebraic Number Theory

The basis of class field theory and Galois cohomology, which are used for deriving almost all the results on p-extensions in this book, is the calculus of cohomology groups of discrete modules of profinite groups. It is this calculus that we have to build first. Here we mainly have to take the requirements for Galois cohomology into consideration, whereas the cohomology of finite groups that is needed for class field theory is only developed as far as is necessary for the formulation of the theorems that will be important for us. See also J.P. Serre [51, Chap. VI–XI].

The Cohomological Dimension of Gs

One of the most important methods for constructing pro-p groups is the presentation by generators and relations. In particular, we shall use this method to present the Galois groups of p-extensions.