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Module MA3419: Galois Theory
- Credit weighting (ECTS)
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5 credits
- Semester/term taught
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Michaelmas term 2015-16
- Contact Hours
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11 weeks, 3 lectures including tutorials per week
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- Lecturer
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Prof Vladimir Dotsenko
- Learning Outcomes
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On successful completion of this module, students will be able to:
- State and explain relationships between properties of field extensions and properties of their automorphism groups.
- Construct explicitly finite fields of low orders.
- Determine Galois groups of polynomials of low degree
- Illustrate on specific examples applications of Galois theory.
- Module Content
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Galois theory demonstrates how to use symmetries of objects to learn something new about properties of those objects, on the example of polynomial equations in one variable with coefficients in a field, and specifically roots of those equations. Students taking the module will see how basics of group theory can be used for solving problems outside group theory, in particular for proving a celebrated result of Abel on non-existence of formulas to solve equations of degree 5 using only arithmetic operations and extracting roots.
- Recollection of relevant results in on groups, fields, and rings. Polynomial rings: UFD/PID property, Gauss lemma, Einstein's criterion.
- Algebraic field extensions. Tower Law, ruler and compass constructions.
- Splitting fields, and their properties. Classification of finite fields.
- Normal and separable extensions. The Primitive Element Theorem. Galois extensions. The Galois correspondence. The Fundamental Theorem of Algebra via Galois Theory
- Algorithm for computing the Galois group of a given polynomial.
- Solubility by radicals. Cyclic, Abelian, solvable field extensions. Abel theorem on equations of degree five.
- Abelian and cyclotomic extensions. Towards Kronecker-Weber Theorem
- Module Prerequisite
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Introduction to Group Theory (MA1214)
and Fields, Rings and Modules (MA2215).
- Assessment Detail
- This module will be examined in a 2-hour examination in Trinity term. Continuous assessment will contribute 20% to the final grade for the module at the annual examination session. Supplemental exams if required will consist of 100% exam.