MATH3041: GALOIS THEORY III

• Algebra II (MATH2581)

• None.

• None.

• To introduce the way in which the Galois group acts on the fieldextension generated by the roots of a polynomial, and to apply this tosome 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 inradicals, cubics, quartics, 'general polynomial', cyclotomicpolynomials.
• Ruler and Compass Constructions: definition, criterion forconstructability, impossibility of trisecting angle, etc.
• Further Topics.

Subject-specific Knowledge:

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

Subject-specific Skills:

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

Key Skills:

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

Eight written 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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