## Mathematics 306## Algebra II |

- explore the world of algebraic structures with a rigorous and axiomatic approach to the evolving mathematics.
- develop and demonstrate a level of expertise in mathematical reasoning appropriate to a foundational upper-level mathematics course.

The link (pdf) is to a file which was generated from an Excel spreadsheet. It can be displayed by your web browser or by an application such as Adobe Acrobat or Preview.

- Math 306 MWF Spring 2007 (pdf)

Much of our course will center on working through a substantial set of exercises taken from the Fraleigh text. Developing clear, clean, and elegant solutions to those problems, and explaining those solutions to each other, will take up much of our class time. You are then to write up your own versions of those solutions and hand them in as homework. The class discussions represent a community effort and may not extend to all of the exercises for which you are responsible. The homework for this course consists in writing a representation of your own understanding of the solutions to these exercises, whether discussed in class or not. Although the class sessions are collaborative and explorative, writing the homework solutions is an individual effort and is definitive. The goal is to engage and internalize the central mathematical issues of correctness, precision of expression, elegance, and style.

The following Mathematica notebooks are available in three formats. Download the Mathematica notebook (nb) to your machine and use Mathematica to interpret its contents, or click on the pdf link (pdf) or the web page link (html) to see a static image of the evaluated notebook.

- Biography and portraits of Joseph-Louis Lagrange (1736-1813). Lagrange first considered permutations of the roots of equations.
- Biography and photos of Arthur Cayley (1821-1895). Cayley introduced Cayley digraphs.
- Biography and photos of Leopold Kronecker (1823-1891). Kronecker proved the structure theorem for finite abelian groups (subsequently extended to the fundamental theorem of finitely generated abelian groups).
- Wikipedia entry on frieze groups. Each frieze group is the symmetry group of one of seven frieze patterns, but up to isomorphism, there are only four frieze groups, two abelian and two non-abelian.
- Colorful Wikipedia entry on wallpaper groups and illustrated summary of the planar symmetry groups. Planar symmetries figure prominently in the work of graphic artist M. C. Escher (1898-1972). This engaging biography of Escher reports that he was strongly influenced by Moorish art, Polya's work on symmetry groups, and his long friendship with British-Canadian mathematician Harold Scott MacDonald Coxeter (1907-2003).
- Biography and sketches of Evariste Galois (1811-1832). Evariste Galois's story is one of the most remarkable biographies in mathematics, and his creation, known today as Galois theory, is widely regarded as one of the most elegant theories in mathematics.
- Wikipedia entry for Burnside's lemma, also known as the orbit-counting theorem. Burnside's lemma was actually discovered by Cauchy 52 years before Burnside's treatment popularized it.
- Biography and photo of David Hilbert (1862-1943). Hilbert introduced the term "ring" to denote the structure of the integers of an algebraic number field.
- Biography and photo of Emmy Noether (1882-1935). Emmy Noether's pioneering work on ideal theory helped to establish the emerging field of commutative algebra. She eventually moved to Bryn Mawr College.
- Biography and sketch of Ferdinand Gotthold Max Eisenstein (1823-1852). Eisenstein was poor and of poor health for much of his short life, but he did receive some modest support from the king and government of Prussia thanks to the generous intervention of the famous explorer Baron von Humboldt.
- Biography and photos of Max August Zorn (1906-1993). Zorn attended the University of Hamburg, and was Emil Artin's second doctoral student, distinguishing himself with prize-winning work on alternative algebras. He immigrated to the United States in 1933, and the following year become a Sterling Fellow at Yale University at the age of 28. A short paper published a year later established what has since become known as Zorn's Lemma. The name "Zorn's Lemma" was conferred on his result by John Tukey.
- Biography and photos of Peter Ludwig Mejdell Sylow (1832-1918). Sylow proved one of the most fundamental results in the theory of finite groups in a 10-page paper published in 1872, when he was 40 years old. For most of his life, Sylow was a high school teacher in the town of Frederikshald, Norway. The enormously influential mathematician, and fellow Norwegian, Sophus Lie created a special position for Sylow at Christiania University in Oslo in 1898. Sylow occupied the position, becoming a university professor at age 66, but Sophus Lie died the following year.
- Wikipedia entries for homology, cohomology, and homology theory. To pursue these topics, see the online text Algebraic Topology by Allen Hatcher of Cornell University.
- A ring that satisfies the ascending chain condition on ideals is a Noetherian ring, named after Emmy Noether (1898-1962; photos). A ring that satisfies the descending chain condition on ideals is an Artinian ring, named after Emil Artin (1898-1962; photos). The Akizuki-Hopkins-Levitzski Theorem shows that Artinian rings are Noetherian. This glossary of ring theory helps to organize the language of ring theory.
- Every compact surface is homeomorphic to a sphere, to an n-handled sphere, or to a sphere with q crosscaps. A crosscap is a projective plane by another name. This impressive classification theorem was first proved in 1907 by Max Dehn (1878-1952; photos) and Poul Heegaard (1871-1948; photos). Max Dehn has a most remarkable personal story. Forced by circumstances to become an itinerant mathematician, he travelled from Germany, through Norway, Siberia, and Japan, to eventually arrive in the United States, and after several short-lived teaching engagements in various American universities he eventually settled into a quiet life working for $40 per month as the only mathematician at Black Mountain College, near Asheville, North Carolina. Some years ago, Steven Puckette, a mathematician and former Dean of the University of the South, took me to see his humble grave, marked only by a small piece of tile among the rhododendrons near Black Mountain. His remarkable story is poignantly told by John Dawson in the article Max Dehn, Kurt Gödel, and the Trans-Siberian Escape Route, published by the American Mathematical Society.
- There are several stepping stones to Galois theory. Evariste Galois (1811-1832; sketches) sets the stage by investigating the solvability of equations by radicals. Julius Wilhelm Richard Dedekind (1831-1916; photos) moves the discussion from permutations of roots of polynomials to mappings of fields, and finally Emil Artin (1898-1962; photos) establishes Galois theory's fundamental correspondence between extension fields and groups of automorphisms in a lecture in 1926. Artin's approach was popularized in B. L. van der Waerden's (1903-1996; photos) influential text
**Modern Algebra**, published in 1930, and, a few years later, by Artin's own notes on the subject. - The biography of Niels Henrik Abel (1802-1829; sketches and statue) is one of the most dramatic in mathematics. He overcame great obstacles to produce truly great mathematics. The proof of Abel's theorem on the insolvability of the quintic uses much of the algebra we have developed over the course of this semester, and is exemplary of the beauty and elegance of modern algebra.

Sewanee's tradition of cordial and constructive student-faculty relationships is one of its great strengths, and I am very happy to support that tradition. If you would like to talk with me, please make an appointment, either when you see me in class or in the hallways or in my office (WL120), or by email (cparrish@sewanee.edu), or by voice mail message (x1333).

Homework is due on Fridays!

- 3 review examinations 60 %
- homework and such 20 %
- final examination 20 %