General information

Module title Galois Theory with Applications
Language English
Module lecturer prof. Wojciech. J. Gajda
Lecturer's email gajda@amu.edu.pl
Lecturer position profesor zwyczajny
Faculty Faculty of Mathematics and Computer Science
Semester 2021/2022 (summer)
Duration 60
ECTS 6
USOS code 06-DTGZUM0-E

Timetable

Module aim (aims)

Much of modern algebra centred around a search for explicit formulae for roots of polynomial equations. The solution of linear and quadratic equations in a single unknown was understood in the antiquity, while formulae for the roots of general real cubics and quartics was solved by the 16th century. By the early 19th century no general solution of a general polynomial equation ‘by radicals’ was know despite considerable effort by many outstanding mathematicians. Eventually, the work of Ruffini, Abel and Galois around 1830 led to a satisfactory understanding of the problem and to the proof that the general polynomial equation of degree at least 5 could not always be solved by radicals. At a deeper level, the algebraic structure of Galois extensions is mirrored in the subgroups of their Galois groups, which allows the application of group theoretic methods to the study of fields. The Galois correspondence is a very powerful tool and can be generalized to such diverse areas as ring theory, algebraic number theory, algebraic geometry, differential equations and algebraic topology. Because of this, Galois theory in its many manifestations is a central topic in modern mathematics. The major aim of the course is to provide students with basic facts and skills of Galois theory which in such a beautiful manner unified algebra (and in a sense also geometry) in the search for a solution of the very natural mathematical problem of solving polynomial equations by algebraic operations.

Pre-requisites in terms of knowledge, skills and social competences (where relevant)

Some basic knowledge of algebra (groups, rings & fields as in a first course on the subject) will be very helpful.

Syllabus

Week 1 Integral domains, fields and polynomial rings IWeek 2 Integral domains, fields and polynomial rings IIWeek 3 Fields extensions, geometric constructionsWeek 4 Algebraic extensions of fields Week 5 Primitive Element Theorem, Normal extensions and splitting fields Week 6 Galois extensionsWeek 7 Galois Correspondence IWeek 8 Galois correspondence IIWeek 9 Galois extensions in positive characteristicWeek 10 Traces and normsWeek 11 Fundamental Theorem of Algebra, computing Galois groupsWeek 12 Cyclotomic fieldsWeek 13 Simple radical extensionsWeek 14 Solvability and radical extensionsWeek 15 Solvability of the quintic, information about elliptic functions

Reading list

• I.Steward, “Galois Theory”, Chapmann & Hall (2003)• D.Cox, “Galois Theory”, Addison-Wesley (2000)• J.-P. Escofier, “Galois theory”, Springer Verlag (2000)• M.Artin, “Galois Theory”, Duke Univ. (1964).