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

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...

18.785F17 Number Theory I Lecture 16 Notes: Riemann’s Zeta Function and the Prime Number Theorem

We now divert our attention from algebraic number theory to talk about zeta functions and L-functions. As we shall see, every global field has a zeta function that is intimately related to the distribution of its primes. We begin with the zeta function of the rational field Q, which we will use to prove the prime number theorem. We will need some basic results from complex analysis, all of whic...

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

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 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 ...