University of Durham --- Department of Mathematical Sciences

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Project IV topics 2021-22

Local class field theory: extensions of local fields with abelian Galois groups.

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.

These solutions can be also expressed in terms of congruences modulo powers of a prime number p, and the coefficients of our equations can be treated as special cases of formal power series in "variable" p with coeffcients 0. 1, ..., p-1. Such elements 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 formal Laurent series Fp((t)). The study of both types of fields (so-called local fieldes) can be done in parallel; this is an essential part of modern Algebraic Number theory.

The topic includes the study of main principles and results 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 relatively simple and contains many interesting algebraic explicit constructions; and
  • symbols in local class field theory, i.e. explicit formulas giving the action of the Galois group on abelian field extensions.

    Prerequisites:

    Galois Theory III, Algebra and Number Theory II

    Resources:

    Local class field theory, by K.Iwasawa

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

    Galois Theory of p-extensions, by H.Koch

    Local fields and their extensiions, by I.Fesenko, S.Vostokov

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