GALOIS THEORY (code: 451)
CREDIT RATING: 10
SEMESTER: SECOND
CONTACT: DR T. VORONOV (MSS/P5)
Aims: To introduce students to a sophisticated mathematical subject where elements
of different branches of mathematics are brought together for the purpose of solving
an important classical problem.
Objectives: On successful completion of the course students will
- have deepened their knowledge about fields
- have acquired sound understanding of the Galois correspondence between
intermediate fields and subgroups of the Galois group
- be able to compute the Galois correspondence in a number of simple examples
- appreciate the insolubility of polynomial equations by radicals.
Pre-requisites: 212, 252, 312
Dependent Courses: None
Course Description: Galois theory is one of the most spectacular mathematical
theories. It gives a beautiful connection of the theory of polynomial equations and
group theory. In fact, many fundamental notions of group theory originated in the
work of Galois. For example, why some groups are called "soluble" ? Because they
correspond to the equations which can be solved ! (Meaning by a solution some
formula based on the coefficients and involving algebraic operations and extracting
roots of various degrees.) Galois theory explains why we can solve quadratic, cubic
and quartic equations, but no similar formulae exist for equations of degree greater
than 4. In modern exposition, Galois theory deals with "field extensions", and the
central topic is the "Galois correspondence" between extensions and groups. For those
familiar with topology, it can bring in mind the relation between coverings and
fundamental group.
Teaching Mode:
2 lectures per week
1 tutorial per week
Private Study:
4 hours per week
Recommended Texts:
I Stewart, Galois Theory, 2nd edition, Chapman and Hall
J B Fraleigh, A first course in abstract algebra, 5th
edition, Addison-Wesley
Assessment Methods: A 2 hour examination at the end of the SECOND Semester.
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Course 451
No of lectures
Syllabus
3
Recollection: factorization of polynomials, irreducibility,
Gauss's Lemma, Eisenstein's test. Zeros and factorization.
Formal derivatives.
3
Field extensions. Simple extensions, algebraic and
transcendental case. Minimum polynomial. Construction of
simple algebraic extension from an irreducible polynomial.
Classification of simple extensions.
3
Degree of extension. Recollection of vector spaces and
dimension. Tower law. Algebraic elements and algebraic
extensions; finite extensions. Algebraic numbers. Geometric
constructions with ruler and compasses.
2
The Galois group of an extension. Examples. The Galois
correspondence between subgroups and intermediate fields.
3
Splitting field for a polynomial. Existence, uniqueness up to
isomorphism. Normal extensions. Relation to splitting fields.
Normality of intermediate extension. Normal closure.
2
Separability. Example of inseparable polynomial. Separability
of all polynomials in characteristic zero. Separable extensions.
Separability of intermediate extensions.
2
Dedekind's Lemma about linear independence of distinct
monomorphisms. Degree of the extension corresponding to a
group of field automorphisms.
3
Normality and monomorphisms. Transitivity of the Galois
group on the zeros of an irreducible polynomial in a normal
extension. Properties equivalent to normality.
3
Galois groups of normal separable extensions. Properties of
Galois correspondence for normal separable extensions.
Normal subgroups and normal intermediate extensions. The
Fundamental Theorem of Galois Theory.
3
Application to polynomial equations. The Galois group of a
polynomial. Galois group as a group of permutations. Solution
by radicals and radical extensions. "Solubility" of the Galois
group of a radical extension. Main properties of soluble groups.
Example of insoluble quintic. "General" polynomial equations.
Revised : January 1999
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