# MA3D5 - Galois Theory

**Module code:**MA3D5**Module name:**Galois Theory**Department:**Mathematics Institute**Credit:**15

Content and teaching | Assessment | Availability

## Module content and teaching

###### Principal aims

The course will discuss the problem of solutions of polynomial equations both in explicit terms and in terms of abstract algebraic structures. The course demonstrates the tools of abstract algebra (linear algebra, group theory, rings and ideals) as applied to a meaningful problem.

###### Principal learning outcomes

By the end of the module the student should understand: 1. The relation between roots and coefficients of a polynomial: elementary symmetric functions; complex roots of unity; and solutions by radicals of cubic and quartic equations; 2. The characteristic of a field and the prime subfield; 3. Factorisation and ideal theory in the polynomial ring $k[x]$ ; the structure of a primitive field extension; 4. Field extensions and characterisation of finite normal extensions as splitting fields; 5. The structure and construction of finite fields; 6. Counting field homomorphisms; the Galois group and the Galois correspondence; 7. Radical field extensions; 8. Soluble groups and solubility by radicals of equations.

###### Departmental link

http://go.warwick.ac.uk/MA3D5/

###### Other essential notes

Prerequisites: MA106 Linear Algebra, MA245 Algebra II.

## Module assessment

Assessment group | Assessment name | Percentage |
---|---|---|

15 CATS (Module code: MA3D5-15) | ||

D (Assessed/examined work) | Assessed worksheets and tests | 15% |

3 hour examination (April) | 85% |

## Module availability

This module is available on the following courses:

###### Core

N/A

###### Optional Core

- Undergraduate Mathematics and Philosophy (GV17) - Year 3
- Undergraduate Mathematics and Philosophy with Specialism in Logic and Foundations (GV19) - Year 4

###### Optional

- Undergraduate Mathematics and Physics (BSc MMathPhys) (FG33) - Year 3
- Undergraduate Mathematics (BSc) (G100) - Year 3
- Undergraduate Mathematics with Intercalated Year (G101) - Year 4
- Undergraduate Mathematics (MMath) (G103) - Year 3
- Undergraduate Mathematics (MMath) (G103) - Year 4
- Undergraduate Master of Mathematics (with Intercalated Year) (G105) - Year 3
- Undergraduate Master of Mathematics (with Intercalated Year) (G105) - Year 4
- Undergraduate Master of Mathematics (with Intercalated Year) (G105) - Year 5
- Undergraduate Mathematics (MMath) with Study in Europe (G106) - Year 3
- Undergraduate Mathematics (MMath) with Study in Europe (G106) - Year 4
- Undergraduate Mathematics and Statistics (BSc MMathStat) (G1G3) - Year 3
- Undergraduate Mathematics and Statistics (BSc MMathStat) (G1G3) - Year 4
- Postgraduate Taught Mathematics (G1P0) - Year 2
- Postgraduate Taught Mathematics (Diploma plus MSc) (G1PC) - Year 2
- Postgraduate Taught Interdisciplinary Mathematics (Diploma plus MSc) (G1PD) - Year 2
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (G300) - Year 3
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (G300) - Year 4
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (G30A) - Year 3
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (G30A) - Year 4
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (G30A) - Year 5
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (G30B) - Year 3
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (G30B) - Year 4
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (G30B) - Year 5
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (G30C) - Year 3
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (G30C) - Year 4
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (G30C) - Year 5
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (G30D) - Year 3
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (G30D) - Year 4
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (G30D) - Year 5
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated (G30E) - Year 3
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated (G30E) - Year 4
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated (G30E) - Year 5
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated (G30F) - Year 3
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated (G30F) - Year 4
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated (G30F) - Year 5
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated (G30G) - Year 3
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated (G30G) - Year 4
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated (G30G) - Year 5
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated (G30H) - Year 3
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated (G30H) - Year 4
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated (G30H) - Year 5
- Undergraduate Discrete Mathematics (G4G1) - Year 3
- Undergraduate Discrete Mathematics with Intercalated Year (G4G2) - Year 4
- Undergraduate Discrete Mathematics (G4G3) - Year 3
- Undergraduate Mathematics and Physics (with Intercalated Year) (GF14) - Year 4
- Undergraduate Mathematics and Statistics (BSc) (GG14) - Year 3
- Undergraduate Mathematics and Statistics (with Intercalated Year) (GG17) - Year 4
- Undergraduate Mathematics and Economics (GL11) - Year 3
- Undergraduate Mathematics and Economics (with Intercalated Year) (GL12) - Year 4
- Undergraduate Mathematics and Philosophy with Specialism in Logic and Foundations (GV19) - Year 3
- Undergraduate Mathematics,Operational Research,Statistics and Economics (Y602) - Year 3
- Undergraduate Mathematics,Operational Research,Statistics,Economics (with Intercalated Year) (Y603) - Year 4