Abstract:
In this thesis we give an exposition of John Tate's doctoral dissertation titled `Fourier
Analysis in Number Fields and Hecke's Zeta-Functions'. In this dissertation, Tate
proved the analytic continuation and functional equation for Hecke's -function over
a number eld k using what is now known as harmonic analysis over ad eles. In his
work he rst examines the local -function and then uses ad eles and id eles to include
in a symmetric way all the completions of the eld into a single structure, so as to
examine the global -function.
We explain required prerequisites and expand upon ideas used in Tate's thesis to
give a comprehensive view of Tate's work.
