# 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.

### Announcements

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.

### Links

- Assignments: html
- Outline of topics covered: html
- Bibliography: html

### Description

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

- the law of quadratic reciprocity
- the study of binary quadratic forms
- cyclotomy and roots of unity
- the unsolvability of the quintic

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.