Galois Theory
Galoisteori

FMAP25, 7.5 credits, A (Second Cycle)

Valid for: 2025/26
Faculty: Faculty of Engineering LTH
Decided by: PLED F/Pi
Date of Decision: 2025-04-10
Effective: 2025-05-05

General Information

Depth of study relative to the degree requirements: Second cycle, in-depth level of the course cannot be classified
Elective for: F5, F5-mtm, Pi5-mtm
Language of instruction: The course will be given in English

Aim

Galois Theory was originally developed by Évariste Galois to characterize the polynomial equations that may be solved with root extractions. It has later also been used to solve classical geometric problems, e.g to show that it is impossible to divide an angle into three equal parts using only a straight edge and a compass.

The course aims to provide a deeper understanding of field extensions, and a connection between the theory of polynomial equations and group theory.

Learning outcomes

Knowledge and understanding
For a passing grade the student must

• be able to give a detailed account of the concepts, theorems and methods included in the course,
• be able to identify the most important theorems in the course and present their proofs.

Competences and skills
For a passing grade the student must

• be able to, in connection with problem solving, integrate knowledge from different parts of the course,
• be able to independently identify problems that can be solved by methods that are part of the course and use appropriate solution methods,
• be able to explain the solution to a mathematical problem within the course framework, in speech and in writing, logically coherent and with adequate terminology.

Judgement and approach
For a passing grade the student must

• be able to argue for the importance of Galois theory as a tool for solving problems in other areas of mathematics, such as the theory of polynomial equations.

Contents

• Field extensions: splitting fields, normal extensions and separable extensions, field automorphisms, normal closures.
• Galois groups: Galois extensions, the Galois Correspondence, the Fundamental Theorem of Galois Theory.
• Polynomial equations: solvability by radicals, insolvable quintics, symmetric polynomials, cyclotomic extension.

Examination details

Grading scale: TH - (U, 3, 4, 5) - (Fail, Three, Four, Five)
Assessment:

The examination consists of a written examination followed by an oral examination at the end of the course. The oral examination may only be taken by those students who passed the written examination.

To pass on the course the student must pass - i.e. have at least 50% of the maximal possible score - on each of the exams. For the grade 4 at least 67% of the joint total are required, and for the grade 5 at least 80% of the joint total.

The maximum number of points that can be obtained on the written and the oral examinations are weighted three to one.

The examiner, in consultation with Disability Support Services, may deviate from the regular form of examination in order to provide a permanently disabled student with a form of examination equivalent to that of a student without a disability.

Modules
Code: 0125. Name: Written Examination.
Credits: 5.0. Grading scale: UG - (U, G). Assessment: Written test comprising theory and problems. The module includes: The whole course.
Code: 0225. Name: Oral Examination.
Credits: 2.5. Grading scale: UG - (U, G). Assessment: Oral examination of the whole course. The module includes: The whole course.

Admission

Admission requirements:

• FMAN10 Algebraic Structures

Reading list

• Stewart, Ian, 1945-: Galois theory [Elektronisk resurs]. Boca Raton, Fla. Chapman & Hall/CRC, 2004, ISBN: 0203489306. Main textbook.
• Cox, David A: Galois Theory [Elektronisk resurs]. Hoboken : John Wiley & Sons, 2012, ISBN: 978-1118072059. Additional recommended reading.
• Rotman, Joseph J., 1934-: Galois theory. New York : Springer, 1998, ISBN: 0387985417. Additional recommended reading.

Contact

Teacher: Anitha Thillaisundaram, anitha.thillaisundaram@math.lu.se
Examinator: Anders Holst, Anders.Holst@math.lth.se
Director of studies: Anders Holst, Anders.Holst@math.lth.se
Director of studies: Anna-Maria Persson, anna-maria.persson@math.lu.se