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

18.785F16 Number Theory I Lecture 26 Notes: Global Class Field Theory, Chebotarev Density

(using the restricted product topology ensures that a 7→ a−1 is continuous, which is not true of the subspace topology). As a topological group, IK is locally compact and Hausdorff. The multiplicative group K× is canonically embedded as a discrete subgroup of IK via the diagonal map x 7→ (x, x, x, . . .), and the idele class group is the quotient CK := IK/K, which is locally compact but not com...

18.785F16 Number Theory I Lecture 18 Notes: Dirichlet L-functions, Primes in Arithmetic Progressions

Notes on (0,2) Superconformal Field Theories

In these lecture notes, I review the " linear σ-model " approach to (0,2) string vacua. My aim is to provide the reader with a toolkit for studying a very broad class of (0,2) superconformal field theories with the requisite properties to be candidate string vacua.

Lectures on Topics in Algebraic Number Theory

2 Preface During December 2000, I gave a course of ten lectures on Algebraic Number Theory at the University of Kiel in Germany. These lectures were aimed at giving a rapid introduction to some basic aspects of Algebraic Number Theory with as few prerequisites as possible. I had also hoped to cover some parts of Algebraic Geometry based on the idea, which goes back to Dedekind, that algebraic n...

18.785F17 Number Theory I Lecture 21 Notes: Class Field Theory

In the previous lecture 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 now want make the correspondence between H and L explicit, and then g...