## Mathematics 305## Algebra I |

- 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 305 MWF Fall 2006 (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.

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 %