46143 - Higher Algebra II

Learning outcomes

The aim of the course is to give an elementary introduction to Galois Theory, with historical notes on the theory of algebraic equations and with attention to computational aspects.

Course contents

Some topics from previous courses in Algebra: polynomials (roots, factorization in irreducibles, criteria of irreducibility) and polynomial equations of degree 3 and 4. Fields extensions, simple extensions, algebraic extensions (degree of an extension, minimal polynomial, splitting fields). Normal extensions. Separable extensions. The Theorem of the Primitive Element. Galois group of an extension. Galois groups of splitting fields. Examples of Galois groups.The Galois correspondence. Splitting fields and separable polynomials, finite separable extensions; normal subgroups and normal extensions, The Fundamental Theorem of Galois Theory. Solvability by radicals of algebraic equations: solvable groups, radical and solvable extensions; solvable extensions and solvable groups; simple groups; solving polynomial equations by radicals. Cyclotomic extensions: cyclotomic polynomials; the Galois group of a cyclotomic extension. inite fields: existence and uniqueness, Galois groups. Geometric constructions using straightedge and compass: constructible numbers, construction of regular polygons. Computation of Galois groups of polynomials.

Readings/Bibliography

A.Caranti - S.Mattarei: Note del corso di Teoria di Galois A.A.2004/2005, Trento (disponibili sul sito del prof. Caranti) D.Cox: Galois Theory Wiley& Son Inc, Hoboken 2004 I. S.Gabelli: Elementi di teoria dei campi Roma III, A.A: 2004/2005 (disponibili sul sito della prof. Gabelli) S.Gabelli: La Corrispondenza di Galois e alcune sue applicazioni, Roma III, A:A: 2005/2006 (disponibili sul sito della prof. Gabelli) Stewart, Galois Theory, Third Edition, Chapman &Hall/CRC, Boca Raton, FL, 2003 J.P.Tignol: Galois'Theory of Algebraic Equations World Scientific, Singapore 2001

Teaching methods

Lectures – exercise sessions - students' seminars – office hours

Assessment methods

Written and oral exam

Teaching tools

Lectures with exercise sessions. Sheets of exercises will be handed out during the lectures, in addition to the ones available in the suggested textbooks. The exercises proposed in these sheets must be solved by the students in written form at least a week before the exam. Students could be asked to prepare a talk on some topics connected with the course. In the office hours students will be coached individually.

Links to further information

http://www.dm.unibo.it/~manaresi

Office hours

See the website of Mirella Manaresi

See the website of Marta Morigi