![]() | Department of Mathematics | |||
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Credits: 20 | Convenor: Dr. N. Snashall | Semester: 1 (weeks 1 to 12) |
Prerequisites: | desirable: MA2102, MA2111 | |
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 |
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.
Algebraic use of polynomials including factorisation.
Computation in various fields.
Written presentation of algebraic arguments in a coherent and logical form.
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.
Know evaluation map is a homomorphism.
State and prove remainder theorem and factor theorem.
Prove that a polynomial of degree has at most
roots in
.
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
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
is maximal iff
is irreducible (in
).
Compute in
Z
, where
is irreducible over Z
, 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 gives
.
Define minimal polynomial and understand its relationship to other
polynomials with the given root.
Prove that minimal polynomial is irreducible.
Understand why
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
and know process for finding an
-basis of
.
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
.
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
in algebraic closure of Z
.
Know that fields of order
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.
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.
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Author: C. D. Coman, tel: +44 (0)116 252 3902
Last updated: 2004-02-21
MCS Web Maintainer
This document has been approved by the Head of Department.
© University of Leicester.