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Durham University

Faculty Handbook 2022-2023

Module Description

Please ensure you check the module availability box for each module outline, as not all modules will run each academic year.

Department: Mathematical Sciences

MATH3041: GALOIS THEORY III

Type Open Level 3 Credits 20 Availability Available in 2022/23 Module Cap Location Durham

Prerequisites

  • Algebra II (MATH2581)

Corequisites

  • None.

Excluded Combination of Modules

  • None.

Aims

  • 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.

Content

  • 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.

Learning Outcomes

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 with minimal guidance: Abstract reasoning.
Key Skills:

    Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

    • 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.

    Teaching Methods and Learning Hours

    Activity Number Frequency Duration Total/Hours
    Lectures 42 2 per week for 20 weeks and 2 in term 3 1 Hour 42
    Problems Classes 8 Four in each of terms 1 and 2 1 Hour 8
    Preparation and Reading 150
    Total 200

    Summative Assessment

    Component: Examination Component Weighting: 100%
    Element Length / duration Element Weighting Resit Opportunity
    Written examination 3 Hours 100%

    Formative Assessment:

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


    Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University



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