MATH30420: Galois Theory

• Prior knowledge of Algebra at undergraduate level.

• None

• None

• To introduce the way in which the Galois group acts on the field extension generated by the roots of a polynomial, and to apply this to some classical ruler-and-compass problems as well as elucidating the structure of the field extension.

• Field Extensions: Algebraic and transcendental extensions, splitting field for a polynomial, normality, separability.
• Results from Group Theory: Normal subgroups, quotients, soluble groups, isomorphism theorems.
• Groups acting on fields: Dedekind's lemma, fixed field, Galois group of a finite extension, definition of Galois extension, fundamental theorem of Galois theory.
• Galois Group of Polynomials: Criterion for solubility in radicals, cubics, quartics, 'general polynomial', cyclotomic polynomials.
• Ruler and Compass Constructions: definition, criterion for constructability, impossibility of trisecting angle, etc.
• Further Topics.

Subject-specific Knowledge:

• By the end of the module students will: be able to solve novel and/or complex problems in Galois Theory.
• have a systematic and coherent understanding of theoretical mathematics in the field of Galois Theory.
• have acquired a coherent body of knowledge of these subjects demonstrated through one or more of the following topic areas: Algebraic field extensions, properties of normality and separability.
• Properties of Galois correspondence.
• Criterion of solvability of polynomial equation in radicals.
• Non-solvability of general polynomial equation in degrees > 5.
• Classification of finite fields.
• Construction of irreducible polynomials with coefficients in finite fields.

Subject-specific Skills:

• In addition students will have specialised mathematical skills in the following areas which can be used in minimal guidance: Abstract reasoning.

Key Skills:

• Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
• Assignments for self-study develop problem-solving skills and enable students to test and develop their knowledge and understanding.
• Formatively assessed assignments provide practice in the application of logic and high level of rigour as well as feedback for the students and the lecturer on students' progress.
• The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

Eight written or electronic assignments to be assessed and returned. Other assignments are set for self-study and complete solutions are made available to students.

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