## Math 225   Modern Algebra

### Prof. D Joyce

#### Department of Mathematics and Computer ScienceClark University, Fall 2008

`[This course page is obsolete.  I'll make another when I teach the course again.]`
• Instructor: David Joyce
Carlson Hall 125. Phone: 793-7421

• Clark University Academic Catalog, Math 225

In the 19th century, Kummer introduced "ideal numbers" to salvage unique factorization of integers into primes (which breaks down in some rings of algebraic integers). This course discusses unique factorization and the modern theory of rings and their ideals, emphasizing Euclidean domains. Other algebraic structures (groups, fields) also are introduced. Required for all mathematics majors. Prerequisite: Math 130 Linear Algebra.

• Course goals.
• To provide students with a good understanding of the theory of modern algebra as described in the syllabus.
• To help students develop the ability to prove theorems and solve problems.
• To introduce students to some of the basic methods of modern algebra.
• To develop abstract and critical reasoning by studying logical proofs and the axiomatic method as applied to modern algebra.
• To make connections between modern algebra and other branches of mathematics, and to see some of the history of the subject.

• Text: no text is required. Course notes will be on line.

• Syllabus. Some details yet to be determined
• Algebraic structures: introduction to fields, rings and groups
• Operations, commutativity, associativity, identity elements, inverse elements, distributivity
• Fields: definition and examples. The fields of real numbers R, of rational numbers Q, and of complex numbers C
• Rings: definition and examples. Commutative and noncommutative rings The ring of integers Z, polynomial rings R[x], matrix rings Mn(R), the ring of integers modulo n Zn
• Groups: definition and examples. Abelian and nonabelian groups The multiplicative group of invertible elements in a ring, groups of symmetries of geometric figures
• Isomorphisms and homomorphisms of rings and groups
• Equivalence relations and congruence
• Congruence modulo n
• Equivalence relations, equivalence classes, partitions, quotient sets
• Cyclic rings Zn
• A little number theory
• Mathematical induction, the well-ordering principal
• Divisibility, greatest common divisor, prime numbers
• The Euclidean algorithm and the extended Euclidean algorithm
• The unique factorization theorem, a.k.a, the fundamental theorem of arithmetic
• Fields
• Formal definition. Subfields.
• F(x) rational functions with coefficients in a given field
• Properties
• Algebraic fields
• Prime fields Zp and other Galois fields, characteristic of a field F
• Ordered fields, complete ordered fields, and R
• Division rings, quaternions H
• Rings
• Formal definition. Commutative rings. Rings with 1. Subrings
• Properties
• Examples besides fields. Z, Zn, matrix rings
• R[x] polynomials with coefficients in a given commutative ring R
• Products of rings
• Chinese remainder theorem
• Boolean rings
• The category of rings, ring homomorphisms, isomorphisms, monomorphisms, etc.
• Integral domains
• Definition. The cancellation law
• Subrings of integral domains and fields. The Gaussian integers
• Finite integral domains are fields
• The field of fractions of an integral domain
• Ideals and quotient rings
• The kernel of a ring homomorphism
• Definition of ideal, examples
• The ideal generated by a subset
• Construction of the quotient ring from an ideal
• Prime and maximal ideals
• Zorn's lemma, Krull's theorem
• Special integral domains: UFDs, PIDs, and EDs
• Divisibility, irreducible elements, prime elements in an integral domain
• Definition and examples of Unique Factorization Domains (UFDs)
• Definition and examples of Principal Ideal Domains (PIDs)
• PIDs are UFDs
• Definition and examples of Euclidean Domains (EDs)
• The division algorithm for polynomials
• Gaussian integers
• EDs are PIDs
• Polynomials
• Basic properties of polynomials
• Complex polynomials and the fundamental theorem of algebra
• Algebraically closed fields
• Real polynomials
• Polynomials with integer and rational coefficients, criteria for irreducibly
• Gauss's lemma, Eisenstein's criterion
• Polynomials with coefficients in a UFD, polynomial rings in several variables
• Introduction to groups
• Definition and examples
• Basic properties of groups and subgroups, cyclic groups
• Cosets, Lagrange's theorem
• Permutations, symmetric groups, alternating groups, dihedral groups
• Cayley's theorem and Cayley graphs
• Linear groups: general, special, orthogonal, unitary, projective
• Abelian groups

• Homework, quizzes, tests, and course grade. There will be weekly homework, a few short quizzes, one or two in-class midterm tests, and a final examination during finals week.

Come to class every day. Expect to put in several hours of work outside of class each week.

• Class notes and other resources

• Pages on the web that you may find interesting.

http://aleph0.clarku.edu/~djoyce/ma225/