Credits: 20 | Convenor: Dr. N.J. Snashall | Semester: 1 |
Prerequisites: | essential: MC241, MC249 | |
Assessment: | Coursework: 10% | Three hour exam in January: 90% |
Lectures: | 36 | Classes: | 10 |
Tutorials: | none | Private Study: | 104 |
Labs: | none | Seminars: | none |
Project: | none | Other: | none |
Total: | 150 |
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.
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.
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 = Zp[x]/(f), where f is irreducible over Zp, 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 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
. 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 Zp.
Know that fields of order pn 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.
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.
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.