Galois extensions and Number Theory

Once one has discovered how to calculate the Galois group of a polynomial, natural questions to ask are "Given a finite group G, is there an extension of Q with Galois group isomorphic to G and can we construct one?". For a general group G, these questions are still open. A lot of work has been done on these and similar problems ('inverse Galois problems'). For some cases the answers come easily: for others the methods employed more sophisticated.

However, these problems have been solved in many cases. One can, for example, write down a recipe for constructing all extensions of Q with given abelian Galois group. The same is true, more or less, for dihedral Galois groups. Some of the methods involved are very interesting (indeed unexpected), some of them using values of elliptic and modular functions.

As well as reviewing and deepening some of last year's work on Galois Theory and Number Theory the project will involve some or all of the following:

• Studying the properties of Galois extensions of algebraic number fields (such as how primes factorize and how their Galois groups are linked with those of extensions of finite fields).
• Studying some of the methods used in the construction of such extensions with desired Galois group or other properties.
• Putting these methods into action. (Some cases can be attacked by `brute force' using the number-theoretic computer languages such as PARI, but any computing needs to be aided by a decent amount of theoretical input.)
• Following up associated interesting sidelines: On the one hand, Kronecker's Jugendtraum (dream of youth) concerns the problem to construct all number fields K with abelian Galois group above a given number field k, using a single transcendental function (for k=Q, the exponential function does the job, while for k imaginary quadratic the elliptic j-function essentially suffices, for other k the problem is wide open)---his is also known as one of the famous Hilbert problems (the 12th one, in fact). On the other hand, one has Stark's conjecture which suggests that values of certain zeta functions for some number field k can be used to find distinguished units in extension fields K and can thereby be used to construct such "class fields".

Prerequisite: Galois Theory III, Number Theory III/IV.

Useful: Elliptic Functions III/IV.

Books:

I.N. Stewart, Galois Theory, Chapman and Hall, ISBN 0412345404

B.L. Van der Waerden, Modern Algebra, Vol. 1, Ungar.

I.N. Stewart and D.O. Tall, Algebraic Number Theory and Fermat's Last Theorem, A.K. Peters, 2001, ISBN 1568811195

(and/or other books from the Galois Theory and Number Theory lists)

Harvey Cohn, Introduction to the Construction of Class Fields, Dover Publications, ISBN-10: 048668346X

Gerald J. Janusz, Algebraic Number Fields, AMS, ISBN-10: 0821804294

David A. Cox, Primes of the form x^2 + n y^2 : Fermat, class field theory, and complex multiplication