On the way to structural mathematics

Nineteenth century algebra and number theory

This is the home page for a seminar taught by Jeremy Avigad (avigad@cmu.edu) and Ken Manders (mandersk+@pitt.edu) in the fall of 2010.


The seminar was offered jointly by Carnegie Mellon University as 80-513/813 and the University of Pittsburgh as Phil 2580 (32878) / HPS 2679 (34636). It met on Fridays from 10-12:20, alternating between Baker Hall 150 (the Philosophy Department Seminar Room) on Carnegie Mellon's campus and Cathedral of Learning 1001-B (the Sellers Seminar Room) on Pitt's campus.



The beginning of the nineteenth century brought striking advances on traditional questions in number theory and the solvability of equations. For example, Gauss proved the law of quadratic reciprocity and gave a detailed analysis of the integers represented by a given quadratic form, and Abel and Galois showed that the general quintic equation has no solution by radicals.

Ongoing work in the nineteenth century was devoted to making sense of these results, and by the end of the century the ideas had been recast in algebraic structural terms. Dedekind, for example, presented Galois theory as the study of field extensions and their automorphisms, and the analysis of quadratic forms in terms of ideals in the ring of algebraic integers of an associated quadratic number field.

By contrasting the earlier and later forms of these results, the course aims to open up a philosophical understanding how modern algebra is so intellectually empowering throughout pure mathematics. We hope to consider

from the early work of Gauss, Abel, and Galois to later, "modern" presentations by Dedekind and Weber.

The course can be taken for either undergraduate or graduate credit, in which case a final grade will be determined based on participation and a written work. Auditors and visitors are welcome.