University of Durham --- Department of Mathematical Sciences

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Project IV topics 2024-25

Galois groups of local fields.

Algebraic number theory concerns algebraic number fields - i.e. field extensions of Q generated by solutions of polynomial equations with integral coefficients. Just as in the case of equations with real coefficients, we can study how many solutions are real or complex, and estimate their absolute values.

Solutions can be expressed in terms of congruences modulo some powers of a prime number p, and the coefficients of our equations are treated as special cases of formal power series in variable p with coeffcients 0. 1, ..., p-1. These form the field of p-adic numbers Qp.

This field has much in common not only with real numbers (its elements appear similarly as infinite decimal fractions) but also with a field of Laurent series Fp((t)). The study of both fields can be done in parallel; this is an essential part of modern Algebraic Number theory.

a. Absolute Galois group of a local field.

This topic includes the study of the theory of local fields, i.e. of finite extensions of either the field of p-adic numbers or the above field of Laurent power series Fp((t)). Any such finite extension contains a unique maximal unramified and tamely ramified subextensions. These subfields can be described in an explicit way and give simplest examples of finite extensions of local fields. More interesting examples come from the study of so-called wildly ramified extensions. One of applications is the proof that any Galois extension of a local field has a soluble Galois group.

Prerequisites:

Galois Theory III, Algebra and Number Theory II

Resources:

Algebraic number theory: proceedings ..., edited by J.W.S.Cassels and A.Frolich

b. Abelian extensions of local fields.

This topic includes the study of local class field theory, which gives a complete description of "abelian" extensions of a given local field K, i.e. Galois extensions L of K such that Gal(L/K) is commutative. In particular, the Galois group of the maximal abelian extension of K is "almost equal" to the group K* of non-zero elements of K, where the operation is induced by multiplication.

Possible directions of development include:

  • the characteristic -p version of this theory, which is very simple and contains many interesting algebraic explicit constructions; and
  • symbols in local class field theory, i.e. explicit formulas which give the action of the Galois group on any given abelian extension.

    Prerequisites:

    Galois Theory III, Algebra and Number Theory II

    Resources:

    Local class field theory, by K.Iwasawa

    c. p-extensions of local fields

    The study of a polynomial equation F(X)=0 with coefficients in a field K leads to the study of all symmetries of the algebraic extension KF of K, which is obtained by joining to K all roots of F(X). This group of symmetries is known as the Galois group of F(X). If we join to K roots of all such polynomials K then we obtain algebraic closure Kalg of K. The Galois group GK of the extension Kalg is called the absolute Galois group of the field K. For example, if K is the field of all complex numbers then GK is the trivial group, but if K is the field of rational numbers then GK is a huge group and we know almost nothing about its structure.

    The study of such groups becomes much easier if we restrict our attention to local fields K (i.e. finite extensions of p-adic fields or fields of Laurent power series) and replace the group GK by its "part" GK(p) coming from extensions of degree equal to a power of this given prime number p. Then the structure of the group GK(p) can be described in terms of generators and relations. For example, if K is the p-adic field then GK(p) has two generators and only one relation.

    The topic includes the study of the simplest properties of the theory of profinite groups and their groups of cohomology.

    Prerequisites:

    Algebra and Number Theory II, Galois Theory III

    Resources:

    Galois Theory of p-extensions, by H.Koch

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