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Department of Mathematics and Applications
"Renato Caccioppoli"


Galois Theory

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Probability and Statistics 2

Credits

6

Prerequisites

Algebra 2.

Scientific-disciplinary sector (SSD)

MAT/02 Algebra.

Examination method

Oral exam.

Learning
objectives

The course aims to develop further critical knowledge of the contents and methods of modern algebra by presenting Galois theory and the elements of group theory and field theory needed to develop it.

Syllabus

Solvable groups, splitting field of a polynomial, n-th roots of unity, Galois fields, normal extensions, algebraically closed fields, separable extensions, perfect fields, Galois group of a field, Galois extensions of a field, fundamental theorem of Galois theory, solvability by radicals, fundamental theorem of algebra.

Expected learning
outcomes

On completion of the course, students must demonstrate that they

  • know and understand the topics covered in group theory, field theory and Galois theory, in particular the issues relating to the solution of algebraic equations;
  • can apply the knowledge acquired to concrete examples and, in particular, compute the Galois group of polynomials of degree three and four;
  • can communicate ideas and solutions clearly, rigorously and effectively to both specialist and non-specialist audiences;
  • can identify the most appropriate methods to analyse and solve a problem relating to the course topics and interpret the results correctly.

Learning outcomes
to be assessed

Command of the knowledge acquired, clarity of presentation, rigour in the use of language, confidence in using the notions acquired.