8430 Handout 5 : Chebotarev Density ; Global Class Field Theory

Remark: This handout presents some of the most important results of algebraic number theory. Although our intended application is to the case of K an imaginary quadratic field, L/K a certain finite abelian extension, R = OK and S = OL, whenever it was not obviously inconvenient I have presented the results in more generality. Some of my motivations for doing this are as follows: first, in contrast to the intended “basic graduate level” audience of Part II of Cox’s book, this course is a topics course which – while, I hope, being mostly accessible to students with a background in basic graduate algebra – is also intended to be useful for students who are now doing or contemplate going on to do thesis work in algebraic number theory and/or arithmetic geometry. For such students – i.e., for at least three out of five – the extra generality I add over Cox’s treatment will almost certainly come in handy later in your career. Moreover, Dino has recently taught two topics courses which develop things in a similar level of generality (and, in fact, with even more loving attention to inseparable field extensions), so it would be a disservice to students who took either or both of his courses not to give the natural continuations of some of the topics from Dino’s courses.