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

ALGEBRAIC FIELD EXTENSIONS
cod. 1005341

Academic year 2015/16
2° year of course - Second semester
Professor
Academic discipline
Algebra (MAT/02)
Field
Attività formative affini o integrative
Type of training activity
Related/supplementary
48 hours
of face-to-face activities
6 credits
hub: PARMA
course unit
in - - -
lectures timetable unavailable
exam calls
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Learning objectives

Good knowledge of the basic aspects of algebraic number theory.

Prerequisites

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Course unit content

Integral extensions: algebraic elements, minimal polynomials, primes in integral extensions, "going up" and "going down" theorems, integrally closed domains.

Dedekind domains: noetherian rings, local Dedekind domains, unique factorization of ideals, class group.

Number fields: finite extensions of the rationals, embeddings in the complex numbers, norm and trace maps, discriminant, ring of integers, examples: quadratic, cubic and cyclotomic fields.

Factorization of primes: factorization in rings of integers, ramification index and inertia degree, Kummer's theorem, Dedekind's theorem, factorization and Galois theory, examples: quadratic and cyclotomic fields.

Full programme

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Bibliography

D.A. Marcus "Number Fields" Universitext, Springer-Verlag.

M.R. Murty - J. Esmonde "Problems in Algebraic Number Theory" GTM 190, Springer-Verlag.

J.S. Milne "Algebraic Number Theory" http://www.jmilne.org/math/CourseNotes/ant.html

Teaching methods

Lectures

Assessment methods and criteria

Oral exam

Other information

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