Complex Number Theory without Imaginary Number (i)

Irrationality Without Number Theory

Complex Cobordism Theory for Number Theorists

The purpose of this paper is to give the algebraic topological background for elliptic cohomology theory and to pose some number theoretic problems suggested by these concepts. Algebraic topologists study functors from the category of spaces to various algebraic categories. In particular there are functors to the category of graded rings called multiplicative generalized cohomology theories. (A...

Analytic Number Theory, Complex Variable and Supercomputers1

This paper is the nal report of an undergraduate honors thesis project advised by Prof. Dennis Hejhal of the School of Mathematics, University of Minnesota. The main purpose of this project is to examine the analytic properties of certain \quantum-mechanical particles" in Lobachevsky space. The results were obtained on vectorized CRAY serial supercomputers, a CRAY 64-CPU T3D massively parallel ...

Class Numbers of Imaginary Abelian Number Fields

Let N be an imaginary abelian number field. We know that hN , the relative class number of N , goes to infinity as fN , the conductor of N , approaches infinity, so that there are only finitely many imaginary abelian number fields with given relative class number. First of all, we have found all imaginary abelian number fields with relative class number one: there are exactly 302 such fields. I...

18.785F16 Number Theory I Lecture 21 Notes: Class Field Theory

In the Lecture 20 we proved the Kronecker-Weber theorem: every abelian extension L of Q lies in a cyclotomic extension Q(ζm)/Q. The isomorphism Gal(Q(ζm)/Q) ' (Z/mZ)× allows us to view Gal(L/Q) as a quotient of (Z/mZ)×. Conversely, for each quotient H of (Z/mZ)×, there is a subfield L of Q(ζm) for which H ' Gal(L/Q). We would like to make the correspondence between H and L explicit, and then ge...