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Università degli Studi di Parma Università degli Studi di Parma
Degree course in
Mathematics
Università degli Studi di Parma
Department of Mathematical, Physical and Computer Sciences

COMMUTATIVE ALGEBRA
cod. 1000143

borrowed from Corso di Laurea in MATEMATICA - ALGEBRA 2 - cod. 1011686
Academic year 2024/25
3° year of course - Second semester
Professor
Andrea APPEL
Academic discipline
Algebra (MAT/02)
Field
A scelta dello studente
Type of training activity
Student's choice
48 hours
of face-to-face activities
6 credits
hub:
course unit
in - - -
lectures timetable
exam calls
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Learning objectives


The students will learn the basic definitions, problems, and techniques in Galois theory.

Prerequisites


Algebra (groups, rings, fields) and Linear Algebra.

Course unit content

Galois theory is one of the most fascinating topics in mathematics, combining elegance and power in a way that few other results can achieve. At its core lies the concept of the symmetry group, used to study polynomial equations and determine when they can be solved by radicals. The fundamental insight is that the structure of the symmetry group (a group of automorphisms associated with the roots of a polynomial) dictates the solvability of the equation. This groundbreaking idea initiated the systematic study of group theory, a cornerstone of modern mathematics. From here emerged Felix Klein’s Erlangen Program, which extended Galois’ ideas to geometry, and the theory of Lie groups, which has become a pillar of contemporary mathematics, especially in algebraic and differential geometry, harmonic analysis, and theoretical physics.

Building on the first year courses of Algebra and Linear Algebra, the course will provide an introduction to the theory of algebraic extensions and Galois theory.

Full programme


Review of group theory, rings, and fields. Classification of finitely generated abelian groups. Structural properties of solvable groups. Finite and algebraic field extensions. Existence theorem of the algebraic closure. Splitting fields, normal extensions. Finite fields. Inseparable extensions. Galois extensions. Cyclic, solvable, and radical extensions. Kummer theory. Galois correspondence theorem. Applications of Galois theory: Galois's proof of the Abel-Ruffini theorem and the constructibility criterion of a complex number. Introduction to algebraic spaces: Nullstellensatz; algebraic spaces and varieties. Spectrum of a ring.

Bibliography


S. Lang, Algebra, Graduate Text in Mathematics, Springer.

Teaching methods


The topics of the course will be discussed during the lectures, together with examples, applications, and numerous exercises.

Assessment methods and criteria


At the end of the course there will be an exam in two parts. The first one amounts to turning in the solutions of a number of exercises. The second one will be at the board, where the student will be asked to discuss, explain, and prove the main results of the course.

Other information

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2030 agenda goals for sustainable development

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Contacts

Toll-free number

800 904 084

Student registry office

E. segreteria.scienze@unipr.it
T. +39 0521 905116

Quality assurance office

Education manager
dott.ssa Giulia Bonamartini
T. +39 0521 906968
E. servizio smfi.didattica@unipr.it
E. del manager giulia.bonamartini@unipr.it

President of the degree course

Prof. Luca Lorenzi
E. luca.lorenzi@unipr.it

Faculty advisor

Prof. Luca Lorenzi
E. luca.lorenzi@unipr.it

Career guidance delegate

Prof. Francesco Morandin
E. francesco.morandin@unipr.it

Tutor Professors

Prof. Emilio Acerbi
E. emilio.acerbi@unipr.it

Prof. Marino Belloni
E. marino.belloni@unipr.it

Prof.ssa Maria Groppi
E. maria.groppi@unipr.it

Prof.ssa Chiara Guardasoni
E. chiara.guardasoni@unipr.it

Prof. Luca Lorenzi
E. luca.lorenzi@unipr.it

Prof. Costantino Medori
E. costantino.medori@unipr.it

Prof. Adriano Tomassini
E. adriano.tomassini@unipr.it

Erasmus delegates

Prof.ssa Fiorenza Morini
E. fiorenza.morini@unipr.it

Quality assurance manager

Prof.ssa Maria Groppi
E. maria.groppi@unipr.it

Tutor students

Dott. Matteo Mezzadri
E. matteo.mezzadri@studenti.unipr.it