Unit APPLIED GALOIS THEORY
- Course
- Mathematics
- Study-unit Code
- A005436
- Curriculum
- Matematica per la crittografia
- Teacher
- Marco Timpanella
- Teachers
-
- Marco Timpanella
- Hours
- 42 ore - Marco Timpanella
- CFU
- 6
- Course Regulation
- Coorte 2025
- Offered
- 2025/26
- Learning activities
- Affine/integrativa
- Area
- Attività formative affini o integrative
- Academic discipline
- MAT/03
- Type of study-unit
- Obbligatorio (Required)
- Type of learning activities
- Attività formativa monodisciplinare
- Language of instruction
- English
- Contents
- To explore the deep connection between group theory and field theory through the framework of Galois theory, culminating in a clear understanding of the Galois Correspondence. To illustrate the power and elegance of this theory by applying it to historically significant problems—such as determining which polynomials have all their roots expressible in terms of radicals—as well as to practical domains like Coding Theory and Cryptography.
- Reference texts
- - D. Cox, Galois Theory, Wiley. - I. Stewart, Galois Theory, CRC Press. - Additional supplementary material available on Unistudium.
- Educational objectives
- The purpose of the course is to: - explain the basic properties of field extensions; - define the notion of Galois group; - show the correspondence between properties of field extensions and properties of the Galois groups attached to them; - apply this correspondence to solve some classical problems of algebra and Euclidean geometry, as well as some more recent objects arising in the context of Coding theory and Cryptography. On completion of this course, the student will be able to: - Demonstrate facility with fields and their extensions, including expertise in explicit calculations with and constructions of examples with various relevant desired properties. - Handle Galois groups, abstractly and in explicit examples, by using a variety of techniques including the Fundamental Theorem of Galois Theory and presentations of fields. - Explain and work with the consequences of Galois Theory in general questions of mathematics addressed in the course, such as insolubility of certain classes of equations or impossibility of certain geometric constructions. - Produce examples and counterexamples illustrating the mathematical concepts presented in the course. - Understand the statements and proofs of important theorems and explain the key steps in proofs, sometimes with variation. The course also aims to refine the students' ability for abstraction and, at the same time, to show how solid theoretical knowledge can lead to the development of meaningful practical tools. Further learning objectives include: Making judgements: To be able to construct and develop logical arguments with a clear identification of assumptions and conclusions; to be able to recognize correct proofs and identify flawed reasoning; to be able to propose interpretations of complex problems within Galois theory and its applications. Communication skills: Ability to present topics, problems, ideas and solutions, both one’s own and others’, in mathematical terms and to communicate conclusions clearly and accurately, using methods appropriate to the intended audience, both orally and in written form. Ability to clearly justify the choice of strategies, methods and content. Learning skills: To read and explore a topic within the field of Galois theory. To independently engage in the systematic study of Algebra topics not previously covered. To equip students with a fundamental tool for further study in Algebra and Number Theory.
- Prerequisites
- To understand and be able to apply most of the techniques covered in the course, it is necessary to have a solid grasp of basic algebraic structures (groups, rings, fields, vector spaces) and to have successfully passed the exams in Algebra I, Algebra II, and Geometry I from the Bachelor's degree in Mathematics.
- Teaching methods
- The course is organized in classroom lectures covering all topics in the syllabus. The lectures are supplemented with notes, exercises, examples, and open research problems. Most of the results will be presented with rigorous proofs, while for some only the statements and their applications will be provided.
- Other information
- Contact the teacher for further information.
- Learning verification modality
- The exam consists of an oral test in which the student will be asked three questions related to three distinct parts of the syllabus. The test lasts approximately 30 to 40 minutes and aims to assess the student’s level of knowledge and understanding of the theoretical and methodological content indicated in the program. The oral exam will also evaluate the student’s communication skills, including the appropriate use of language and the ability to organize and present the theoretical topics independently. For information about support services for students with disabilities and/or specific learning disorders (DSA), please visit the page http://www.unipg.it/disabilita-e-dsa
- Extended program
- Field extensions and their basic properties. The structure and construction of finite fields. Algebraic closure of a field: existence and uniqueness. Kronecker construction. Splitting fields and normal extensions. Separable, inseparable and purely inseparable extensions. Primitive element theorem. Galois Extensions. Galois group and Galois correspondence for finite extensions. Galois group of a polynomial. Cyclical extensions and Kummer theory. Solvable groups. Solvability by radicals, insolvability of quintics. Further examples and applications. Selected other topics may be covered, for example ruler and compass constructions, separability, finite fields.