18.785F17 Number Theory I Lecture 24 Notes: Artin Reciprocity

18.785F17 Number Theory I Lecture 23 Notes: Tate Cohomology

In this lecture we introduce a variant of group cohomology known as Tate cohomology, and we define the Herbrand quotient (a ratio of cardinalities of two Tate cohomology groups), which will play a key role in our proof of Artin reciprocity. We begin with a brief review of group cohomology, restricting our attention to the minimum we need to define the Tate cohomology groups we will use. At a nu...

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...

18.785F17 Number Theory I Lecture 18 Notes: Dirichlet L-functions, Primes in Arithmetic Progressions

18.785F17 Number Theory I Lecture 17 Notes: The Functional Equation

In the previous lecture we proved that the Riemann zeta function ζ(s) has an Euler product and an analytic continuation to the right half-plane Re(s) > 0. In this lecture we complete the picture by deriving a functional equation that relates the values of ζ(s) to those of ζ(1− s). This will then also allow us to extend ζ(s) to a meromorphic function on C that is holomorphic except for a simple ...

18.785F17 Number Theory I Lecture 28 Notes: Global Class Field Theory, Chebotarev Density

the unit group of AK but has a finer topology (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...