20410408 - AL310 - ELEMENTS OF ADVANCED ALGEBRA

Acquire a good knowledge of the concepts and methods of the theory of polynomial equations in one variable. Learn how to apply the techniques and methods of abstract algebra. Understand and apply the fundamental theorem of Galois correspondence to study the "complexity" of a polynomial.

Curriculum

teacher profile | teaching materials

Programme

Fields extensions and their basic properties.

Algebraic closure of a field: existence and uniqueness. Kronecker's 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.

Prefinite groups and Krull topology. Galois correspondence for infinite extensions.

Galois group of an equation. Cyclotomic extensions. Generic equation of degree n.

Linear independence of characters. Trace and norm. Hilbert 90 theorem. Cyclic extensions and Kummer theory.

Solvable groups. Solvable and solvable by radicals extensions.

More examples and applications.

Core Documentation

Algebra S. Bosch

Algebra S. Lang

Algebra M. Artin

Class Field Theory J. Neukirch

Reference Bibliography

Algebra S. Bosch Algebra S. Lang Algebra M. Artin Class Field Theory J. Neukirch

Type of delivery of the course

Lectures in class on blackboard and exercise classes. The students should enroll in the course on Moodle and Teams. The communications will be held through these channels.

Type of evaluation

The exam will consist of a written and an oral exam of the topics studied in the course.

teacher profile | teaching materials

Mutuazione: 20410408 AL310 - ISTITUZIONI DI ALGEBRA SUPERIORE in Matematica L-35 CAPUANO LAURA, TALAMANCA VALERIO

Programme

Fields extensions and their basic properties.

Algebraic closure of a field: existence and uniqueness. Kronecker's 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.

Prefinite groups and Krull topology. Galois correspondence for infinite extensions.

Galois group of an equation. Cyclotomic extensions. Generic equation of degree n.

Linear independence of characters. Trace and norm. Hilbert 90 theorem. Cyclic extensions and Kummer theory.

Solvable groups. Solvable and solvable by radicals extensions.

More examples and applications.

Core Documentation

Algebra S. Bosch

Algebra S. Lang

Algebra M. Artin

Class Field Theory J. Neukirch

Reference Bibliography

Algebra S. Bosch Algebra S. Lang Algebra M. Artin Class Field Theory J. Neukirch

Type of delivery of the course

Lectures in class on blackboard and exercise classes. The students should enroll in the course on Moodle and Teams. The communications will be held through these channels.

Type of evaluation

The exam will consist of a written and an oral exam of the topics studied in the course.

Mutuazione: 20410408 AL310 - ISTITUZIONI DI ALGEBRA SUPERIORE in Matematica L-35 CAPUANO LAURA, TALAMANCA VALERIO