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## ECM3738 - Galois Theory (2018)

MODULE TITLE | Galois Theory | CREDIT VALUE | 15 |
---|---|---|---|

MODULE CODE | ECM3738 | MODULE CONVENER | Prof Nigel Byott (Coordinator) |

DURATION: TERM | 1 | 2 | 3 |
---|---|---|---|

DURATION: WEEKS | 11 | 0 |

Number of Students Taking Module (anticipated) | 30 |
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Drawing on key ideas in the theory of groups and fields, you will learn core elements of the theory of field extensions. You are already familiar with the idea that the real numbers can be extended to the complex numbers by introducing a new number as the square root of -1; Galois theory formalises such constructions and explores the intriguing relationship between groups and field extensions.

As an important application of Galois theory, you will understand why there can be no algebraic solution to the general quintic polynomial with rational coefficients.

Prerequisite module: ECM2711 and ECM2712 or equivalent

The aim of this module is to motivate and develop Galois theory both as an abstract theory and through the study of important applications.

On successful completion of this module, **you should be able to**:

**Module Specific Skills and Knowledge:**

1 state and apply key definitions in Galois theory;

2 state, prove and apply core theorems in Galois theory.

**Discipline Specific Skills and Knowledge**:

3 perform computations accurately;

4 use abstract reasoning to solve a range of problems.

**Personal and Key Transferable / Employment Skills and Knowledge**:

5 communicate your findings effectively in writing;

6 work independently and manage your time and resources effectively.

- review of the field axioms, the characteristic of a field, examples. Field extensions, degree, finite and algebraic extensions, extensions obtained by adjoining a root of an irreducible polynomial, degree in a tower of extensions;

- splitting fields and algebraic closure. Separable and inseparable extensions. Cyclotomic polynomials and extensions. Automorphisms of a field. The group of automorphisms, the fixed field of a subgroup of automorphisms, the Galois correspondence. The fundamental theorem of Galois theory. Finite fields. Finite extensions of finite fields. The Galois theory of finite fields. Composite extensions and simple extensions. The primitive element theorem;

- cyclotomic extensions and abelian extensions over Q. Abelian groups as Galois groups over Q. Cyclic extensions and Kummer theory. Galois groups of polynomials. Solvable and radical extensions: solution of cubic and quartic equations by radicals, insolvability of the quintic. Computation of Galois groups over Q. Hilbert's irreducibility theorem. Polynomials with Galois groups Sn and An.

Scheduled Learning & Teaching Activities | 33.00 | Guided Independent Study | 117.00 | Placement / Study Abroad | 0.00 |
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Category | Hours of study time | Description |

Scheduled learning and teaching activities | 33 | Lectures including example classes |

Guided independent study | 117 | Lecture and assessment preparation; wider reading |

Form of Assessment | Size of Assessment (e.g. duration/length) | ILOs Assessed | Feedback Method |
---|---|---|---|

Exercises | One sheet fortnightly (or equivalent) | All | Verbal and generic feedback in example classes |

Coursework | 20 | Written Exams | 80 | Practical Exams | 0 |
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Form of Assessment | % of Credit | Size of Assessment (e.g. duration/length) | ILOs Assessed | Feedback Method |
---|---|---|---|---|

Written exam – closed book | 80 | 2 hours - Summer Exam Period | All | Exam mark – verbal feedback on request |

Coursework – based on questions submitted for formative assessment | 20 | 30 hours in total | All | Annotated script and written feedback |

Original Form of Assessment | Form of Re-assessment | ILOs Re-assessed | Time Scale for Re-assessment |
---|---|---|---|

All above | Written exam (100%) | All | August Ref/Def period |

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

information that you are expected to consult. Further guidance will be provided by the Module Convener

**ELE: ***College to provide hyperlink to appropriate pages*

Reading list for this module:

Type | Author | Title | Edition | Publisher | Year | ISBN | Search |
---|---|---|---|---|---|---|---|

Set | Stewart, I. | Galois Theory | Chapman and Hall | 2004 | [Library] | ||

Set | Rotman, J. | Galois Theory | Springer | 1998 | [Library] |

CREDIT VALUE | 15 | ECTS VALUE | 7.5 |
---|---|---|---|

PRE-REQUISITE MODULES | ECM2711, ECM2712 |
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CO-REQUISITE MODULES |

NQF LEVEL (FHEQ) | 6 | AVAILABLE AS DISTANCE LEARNING | No |
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ORIGIN DATE | Thursday 06 July 2017 | LAST REVISION DATE | Wednesday 27 February 2019 |

KEY WORDS SEARCH | Galois; field; extension; group; polynomial. |
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