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MC382 Abstract Algebra

MC382 Abstract Algebra

Credits: 20 Convenor: Dr John Watters Semester: 1
Prerequisites: essential: MC241, MC254
Assessment: Coursework: 10% Three hour exam in January: 90%
Lectures: 36 Problem Classes: 10
Tutorials: none Private Study: 104
Labs: none Seminars: none
Project: none Other: none
Surgeries: none Total: 150

Explanation of Pre-requisites

In dealing with one field containing another, e.g. the field of complex numbers containing the real numbers, 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. In considering the structure of fields we also require many of the basic ring properties from MC254.

Course Description

The topics for this module cover three broad areas. The first part considers some basic ring-theoretic notions and enables us to determine when a factor ring is a field. 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 fields, splitting fields and shows how all finite fields can be constructed. It also includes a study of ruler and compass constructions and in particular it is shown that the classical problem of squaring the circle is impossible, that is, it is not possible to construct a square of area equal to that of a given circle.

Aims

The module aims to build on previous study to provide an introduction to fields within the context of a general ring-theoretic framework. 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, the splitting field, can be economically constructed over which the given polynomial factorises into irreducible factors. It is shown how all finite fields can be constructed. Finally, the results proved in this module are applied to ruler and compass constructions.

Objectives

To identify rings and fields.

To relate ideals to factor rings and homomorphisms.

To prove and use tests for irreducibility.

To relate irreducible elements and maximal ideals in the ring of polynomials over a field.

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.

To understand how the concepts introduced in the course may be applied to ruler and compass constructions.

Transferable Skills

Application of taught principles and concepts to new situations.

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

Algebraic use of polynomials including factorisation.

Computation in various fields.

Written presentation of algebraic arguments in a coherent and logical form.

Syllabus

Define the characteristic of a ring and be able to find it in basic examples. Prove that the characteristic of an integral domain is either $0$ or prime. Prove that every field is an integral domain and know an example to disprove converse. Divide polynomials over a field. Recall concept of Euclidean domain with Z, $F[x]$ as examples. Definition of GCD in general domains. Verify subring and subfield properties in elementary examples. Identify subfields of the form Q$[\sqrt d]$. Recall concepts of a ring homomorphism, ideal, right (left) ideal, principal ideal in a commutative ring. Prove that every ideal in $F[x]$ is principal. Define prime and maximal ideals. Describe the ideal structure of the ring of integers. Prove, for a commutative ring $R$, that $I$ is prime iff $R/I$ is an integral domain, and that $I$ is maximal iff $R/I$ is a field.

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 polynomials 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 Q. State and prove Gauss's Lemma, that for degree $f(x) > 0; f(x)$ irreducible over Q iff ``irreducible over Z". State, prove, and use Eisenstein's criterion and Modular Irreducible Test. Decide irreducibility of quartics, quintics over Q and finite fields. Prove that $(f)$ is maximal iff $f$ is irreducible (in $F[x]$). Compute in $E =$ Z$_{p}[x]/(f)$, where $f$ is irreducible over Z$_{p}$, including finding inverses. Define cyclotomic polynomials and prove their irreducibility over Q.

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 minimal polynomial. 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, minimal polynomial 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 $p^n$ in algebraic closure of Z$_{p}$. Know that fields of order $p^n$ are isomorphic. Define a ruler and compass construction. State and prove the condition on field extensions for a point to be constructible. Be able to show the impossibility of squaring the circle.

Reading list

Recommended:

R. B. J. T. Allenby, Rings, Fields and Groups, Arnold.

J. B. Fraleigh, A First Course in Abstract Algebra, 5th edition, Addison-Wesley.

I. N. Herstein, Topics in Algebra, Wiley.

W. K. Nicholson, Introduction to Abstract Algebra, PWS.

I. Stewart, Galois Theory, Chapman and Hall.

Details of Assessment

Coursework - there will be 9 pieces of work set for assessment which are all of equal weight and together count for 10% of the final mark.

Examination - this is of three hours duration and counts for 90% of the final mark. The paper contains eight questions; any number of questions may be attempted, but only the best five answers will be taken into account. Full marks may be obtained for answers to five questions. All questions carry equal weight.

Next: MC383 Complex Analysis Up: Year 3 Previous: MC381 Modelling physical systems

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Last updated: 2001-09-20
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