Galois groups and ramification

Thursday, November 17, 2009

Speaker: Romyar Sharifi

Title: Galois groups and ramification

Abstract: In number theory, one is most concerned with the arithmetic
of finite extensions of the rational numbers, known as number fields.
The arithmetic of such fields is reflected in the Galois groups of its
field extensions of finite degree, these groups acting as symmetries
of roots of polynomials with coefficients in the ground field. One is
therefore interested in knowing which isomorphism classes of Galois
groups can occur for such extensions. Much of the finer structure of
the arithmetic of a number field can be found in objects such as its
class group, the quotient group of nonzero ideals modulo principal
ideals of its ring of integers. In understanding this, the
decomposition of prime ideals of this integer ring in the ring of
integers of extension fields comes into play. For instance, a prime
is said to ramify if some prime of the larger field containing it
occurs to power at least two in the decomposition, this being the
arithmetic analogue of branching in covering spaces in topology. One
might then ask which Galois groups can occur as field extensions that
are ramified at most at these primes. We will discuss a sampling of
what is known in answer to this remarkably deep question, as well as
other questions to which this leads.