MAS6310 Algebra I

 Both semesters, 2019/20 20 Credits Lecturer: Dr Kirill Mackenzie Home page Timetable Aims Outcomes Teaching Methods Assessment Full Syllabus

This module considers Fields and Galois Theory. The first half of the module covers basic field theory (field extensions, constructability, etc.) and the second gives the applications to the theory of equations: Galois groups of extensions, and of polynomials, culminating in one of the crowning glories of Galois’s work – classifying polynomials which are soluble by radicals.

There are no prerequisites for this module.
No other modules have this module as a prerequisite.

Outline syllabus

• Semester 1: Field extensions
• Factorization of polynomials
• Simple field extensions
• Towers of fields
• Ruler and compass constructions
• Semester 2: Review of fields and other background
• Homomorphisms and field extensions
• Splitting fields
• Extending homomorphisms; normal field extensions; Galois groups
• Examples involving extensions of small degree
• Cyclotomic fields and their Galois groups
• Finite fields (if time permits)
• The Galois correspondence
• Cubics and quartics
• Extension by radicals and solvability.

Office hours

Thursdays 10am to 10.50 in J6a

Aims

• introduce basic concepts in field theory;
• introduce the notion of field extension, and study its properties;
• apply these notions to the theory of equations;
• classify equations soluble by radicals;
• explain the definition of Galois groups, and to compute them for cyclotomic extensions, and various extensions of small degree;
• explain the Galois correspondence, and use it to reduce various questions in field theory to easier questions about finite groups.

Learning outcomes

By the end of the unit, a candidate will be able to demonstrate the ability to, the capability for 1. make computations in fields and with field extensions; 2. understand constructability of polygons by ruler and compasses; 3. compute Galois groups of polynomials; 4. show that there are polynomials not soluble by radicals.

Teaching methods

Lectures, problem solving

40 lectures, no tutorials

Assessment

The module will be assessed by a formal, closed book, two and a half hour examination at the end of each semester.

Full syllabus

Semester 1

1. Field extensions
(4 lectures)
Reminders about fields; examples, including Zp when p is a prime number; finite fields; field of rational functions. Subfields, extension fields; subfield criterion. Every subfield of R or C contains Q. The subfield of a field L generated by a subset X of L. Field adjunction; examples; simple field extensions.
2. Factorization of polynomials
(3 lectures)
Reducible polynomials; irreducible polynomials; primitive polynomials in Z[x]; Gauss's Lemma; the important corollary that a primitive polynomial in Z[x] which is irreducible in Z[x] is irreducible in Q[x]. A monic cubic f(x) ∈ Z[x] with no integer root is irreducible over Q. Eisenstein's irreducibility criterion; examples; use of the criterion in conjunction with the fact that, for a non-constant f ∈ Z[x] and k ∈ Z, f(x) is irreducible if and only if f(x+k) is; use of Eisenstein's criterion in conjunction with the fact that f(x) is irreducible if and only if xnf(1/x) is, where n is the degree of f. Primitive n-th roots of unity, where n ∈ N; the cyclotomic polynomial φn; φp is irreducible if p is prime.
3. Simple field extensions
(3 lectures)
Elements of a field extension which are algebraic or transcendental over the base field K; simple algebraic extensions; simple transcendental extensions; minimal polynomial of an algebraic element a, and its irreducibility; if f(a) = 0 for a monic irreducible polynomial f(x) ∈ K[x], then f(x) is the minimal polynomial of a over K; use of the minimal polynomial to express the inverse of a non-zero element of K(a) as g(a) for some polynomial g(x) ∈ K[x].
4. Towers of fields
(3 lectures)
The degree of a field extension; finite extensions; the dimension formula [M:K] = [M:L][L:K] when L is an intermediate field between K and M. The degree of a simple field extension; examples. The set of algebraic numbers is a subfield of C. Splitting fields (of polynomials over subfields of C).
5. Ruler and compass constructions
(7 lectures)
Constructible points of R2; constructible real numbers; examples. The classical problems: doubling the cube; trisecting angles; squaring the circle; constructing regular polygons. Standard constructions. The field of constructible real numbers. Quadratic extensions. Necessary and sufficient (field theoretic) conditions for (a,b) ∈ R2 to be a constructible point. If a ∈ R is constructible, then a is algebraic over Q and [Q(a):Q] is a power of 2. Solutions of the classical problems: statement (no proof, but some history) that π is not algebraic over Q. Constructible complex numbers. Fermat primes. If p is an odd prime which is not a Fermat prime, then the regular p-gon cannot be constructed. Statement of converse, and proof for p = 5, 17. Determination of those integers n ;eq 3 for which the regular n-gon can be constructed.
Semester 2
1. Fields and vector spaces
Rapid review of MAS333/438 and other background.
2. Ideals and quotient rings
Mostly revision.
3. Polynomials over fields
Mostly revision.
Adjoining roots, and properties of the resulting field extensions. Splitting fields.
5. Extending homomorphisms
Dedekind's lemma. Inequalities for numbers of extensions. Normal extensions, and criteria for normality. Galois groups.
6. Some extensions of small degree
Selected examples such as Q(√2,√3) and Q(√{3+√5}).
7. Cyclotomic extensions
Definition and properties of cyclotomic polynomials and their splitting fields. Automorphisms of cyclotomic fields. Examples of subfields of cyclotomic fields.
8. Finite fields
There is a field of each prime-power order, and it is unique up to isomorphism. Galois groups of extensions of finite fields. (May be omitted if time is short.)
9. The Galois correspondence
For any normal extension L/K, there is a bijection between subgroups of the Galois group G(L/K) and intermediate fields between K and L. Finer properties of this bijection. Examples of using the correspondence to derive facts about field structure.
10. Cubics
Galois groups of splitting fields of cubics. The discriminant criterion for whether the group is A3 or Σ3.
11. Quartics
Subgroups of the symmetric group Σ4. Corresponding facts about quartics, including the resolvent cubic.
12. Cyclic extensions
Splitting fields for polynomials of the form xn−a, and their Galois groups.
13. Solvability and extension by radicals
Basic theory of solvable groups. Polynomials are solvable by radicals if and only if the corresponding Galois groups are solvable. Quintics are usually not solvable by radicals.

Timetable (semester 2)

 Mon 14:00 - 14:50 lecture Hicks Seminar Room F20 Fri 10:00 - 10:50 lecture Hicks Seminar Room F38