MC344 Abstract Algebra

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MC344 Abstract Algebra

Credits: 20

Convenor: Dr. J.F. Watters

Semester: 1

Prerequisites:	essential: MC241	desirable: MC242
Assessment:	Coursework: 10%	Three hour exam: 90%

Lectures:	36	Classes:	none
Tutorials:	12	Private Study:	102
Labs:	none	Seminars:	none
Project:	none	Other:	none
Total:	150

Explanation of Pre-requisites

In dealing with one field containing another, e.g the field of comlex numbers containing the reals, we can consider the larger field as a vector space over the smaller and so exploit the idea of basis and other linear algebra concepts to learn about the larger field. Thus MC241 (and its prerequisites) is needed for this course. Many of the basic ring properties have analogues in groups, so MC242 provides useful background.

Course Description

The topics for this module cover three broad areas. The first is a general introduction to rings dealing with basic notions like homomorphisms and ideals. The second section discusses in some detail properties of polynomials over fields, especially over the rationals, and looks particularly at irreducibility. The final part of the module deals with extension field, splitting fields, and shows how all finite fields can be constructed.

Aims

The module aims to provide an introduction to rings and fields. Once the basic ring-theoretic ideas are in place the module develops the concept of irreducibility of polynomials and presents tests for irreducibility. For polynomials in general the module aims to show how a field can be economically constructed, the splitting field, over which the given polynomial factorises into irreducible factors. Finally, it is shown how all finite fields can be constructed.

Objectives

To identify rings and fields.

To relate ideals to factor rings and homomorphisms.

To prove and use tests for irreducibility.

To know the significance and properties of the minimal polynomial.

To know how to establish and use the concept of the degree of an extension.

To construct splitting fields.

To know how to construct finite fields.

Transferable Skills

Developing understanding of the abstract method, ring-theoretic and field-theoretic ideas.

Ability to work algebraically with polynomials including factorisation.

Ability to compute in various fields.

Written presentation of algebraic arguments.

Syllabus

Basic examples. Distinguish commutative and non-commutative rings. Know the definition of a ring. Uniqueness of zero, identity, and negatives. Generalized distributive and associative laws. Property to recognize . Multiplicative properties of and -. Rules for multiples and exponents. Definition and examples of integral domains. Define the characteristic of a ring and find it in basic examples. Prove that an integral domain's characteristic is either or prime. Define units in rings and find them in basic examples. Definition and examples of fields. Prove that every field is an integral domain and know an example to disprove converse.

Divide polynomials over a field. R[x] is a ring; commutative when R is. Degree of product is $\leq$ sum of degrees. Prove division algorithm for polynomials over a field. Concept of Euclidean domain with the integers, F[x] as examples. Definition of GCD in general domains. Understand why Euclidean algorithm works in any ED. Verify subring properties in elementary examples. Identify subfields that are extensions of the rationals by a single square root. Define a ring homomorphism, know elementary properties of these, and special types. Prove that image set is a subring. Prove that one-to-one is equivalent to zero kernel. Define ideal, right (left) ideal. Examples where right and left ideals are different. Define a principal ideal in a commutative ring. Prove that every ideal in F[x] is principal. Describe the ideals in the ring of integers. Prove that every kernel is an ideal. Define cosets in R. Condition for equality of cosets and proof. Cosets are either equal or disjoint. Define ring structure on R/I. Prove that operations on R/I are well-defined. Prove $R/ \mbox{\rm ker\ }\phi \cong im \phi$ . Examples: the integers, and the integers mod n. Define maximal ideals. Know maximal ideals in the ring of integers. Prove I maximal iff R/I field (R commve.).

Know evaluation map is a homomorphism. State and prove remainder theorem and factor theorem. Prove that a polynomial of degree n has at most n roots in F. Define irreducible polynomial. Prove that linear polynomials are irreducible. Prove that polyls. of degree 2 or 3 are reducible iff they have roots. State and prove root test for rational polynomials. Write quadratics and cubics as products of irreducibles over small fields and the rationals. State and prove Gauss's Lemma. deg f(x) > 0;f(x) irred. over the rationals iff ``irred. over the integers". State, prove, and use Eisenstein's criterion and Modular Irreducible Test. Decide irreducibility of quartics, quintics over the rationals and finite fields. Know that (f) is maximal iff f is irreducible (in F[x]). Compute in E = Z_p[x]/(f), f is irreducible over Z_p, including finding inverses. Define cyclotomic polynomials and prove their irreducibility over the rationals.

Define extension fields and degree. Know examples of finite and infinite degree. Be able to prove that every polynomial has a root in some extension. Define algebraic and transcendental elements and know examples. Prove that a transcendental $\alpha$ gives $F[\alpha] \cong F[x]$ . Define minimal polynomial and understand its relationship to other polynomials with the given root. Prove that minimal polynomial is irreducible. Understand why $[F(\alpha) :F] =$ degree of min. poly. Define simple extensions and understand tower construction process. Define an algebraic extension and prove that finite extensions are algebraic. State and prove connections between three degrees when $K \supseteq F \supseteq L$ and know process for finding an L-basis of K. Prove that degree of an element in a simple extension divides the degree of the extension. Be able to use tower construction to obtain basis, degree, min. poly. for finite extensions. Prove that all finite extensions can be obtained as towers of algebraic extensions and vice-versa. Define splitting fields. Find splitting fields and their degrees in simple examples. Prove that a splitting field always exists with degree at most $n\!$ . Know that splitting field is unique up to isomorphism. Define algebraically closed field and algberaic closure of a field. Prove that algebraically closed fields have no proper algebraic extensions. Show that the field of algebraic numbers is algebraically closed. Prove that finite fields must have prime characteristic and prime power order. Be able to compute primitive roots (elements) in finite fields. Know that the multiplicative group of a finite field is cyclic. Define formal derivative of a polynomial. State and prove condition for polynomial to have repeated roots. Construct field of order pⁿ in algebraic closure of the field of integers modulo p. Know that fields of order pⁿ are isomorphic.

Reading list

Details of Assessment

Coursework - ten pieces of work all of equal weight;
Examination - three hours duration, contains eight questions all of equal weight with full marks gained from answers to five. The Casio FX82 is the only calculator allowed in this examination.

Next: MC346 Communicating Mathematics Up: Year 3 Previous: MC342 History of mathematics

Roy L. Crole
10/22/1998