# Matematica | ALGEBRA II

## Matematica ALGEBRA II

 0512300007 DIPARTIMENTO DI MATEMATICA MATHEMATICS 2013/2014

 OBBLIGATORIO YEAR OF COURSE 2 YEAR OF DIDACTIC SYSTEM 2010 PRIMO SEMESTRE
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/02 6 48 LESSONS COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS MAT/02 2 24 EXERCISES COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS
 PATRIZIA LONGOBARDI T MARIA TOTA
Objectives
OUR AIM IS TO CONTINUE THE STUDY, ALREADY STARTED DURING THE COURSE OF ALGEBRA I, OF ALGEBRAIC STRUCTURES, LIKE RINGS AND VECTOR SPACES. IN PARTICULAR FACTORIAL, PRINCIPAL AND EUCLIDEAN DOMAINS WILL BE CONSIDERED. MOREOVER FIELD THEORY WILL BE DEVELOPED TILL SOME FUNDAMENTAL CONCEPTS OF GALOIS THEORY. EXAMPLES AND APPLICATIONS WILL HELP STUDENTS TO BE ACQUAINTED TO THESE THEORIES, TO THEIR TECHNIQUES, TO THEIR MOTIVATIONS, ALSO IN VIEW OF POSSIBLE FUTURE DEVELOPMENTS.
Prerequisites
GOOD KNOWLEDGE OF THE SUBJECTS CONTAINED IN THE COURSE OF ALGEBRA I.
Contents
THE THEORY OF RINGS

NILPOTENT, IDEMPOTENT ELEMENTS. THE FIELD OF QUOTIENTS. UNIQUE FACTORISATION DOMAINS. PRINCIPAL DOMAINS. EUCLIDEAN DOMAINS. NOETHERIAN AND ARTINIAN DOMAINS.

POLYNOMIALS

THE UNIVERSAL PROPERTY. GAUSS’ LEMMA. THE EISENSTEIN CRITERIUM. ROOTS OF A POLYNOMIAL. THE FUNDAMENTAL POLYNOMIAL OF A FINITE FIELD. THE HILBERT BASIS THEOREM.

VECTOR SPACES

ISOMORPHIC VECTOR SPACES. DECOMPOSITIONS. FINITE DIMENSIONAL VECTOR SPACES. EXISTENCE OF VECTOR SPACES OF ARBITRARY DIMENSION. THE ADDITIVE STRUCTURE OF A DIVISION RING.

THE THEORY OF FIELDS

ALGEBRAIC AND TRANSCENDENTAL ELEMENTS. ALGEBRAIC EXTENSIONS. THE ALGEBRAIC CLOSURE OF A SUBFIELD IN A FIELD. CANTOR’S THEOREM. SPLITTING FIELDS. FUNDAMENTAL THEOREMS. ROOTS OF UNITY. FINITE FIELDS. ALGEBRICALLY CLOSED FIELDS. THE ALGEBRAIC CLOSURE OF A FIELD.

GALOIS THEORY

THE GALOIS GROUP OF AN EXTENSION. THE FIXED FIELD OF A GROUP OF AUTOMORPHISMS OF A FIELD. THE GALOIS GROUP OF A POLYNOMIAL. ELEMENTS OF GALOIS THEORY.
Teaching Methods
LECTURES.
EXERCISES.
ATTENDANCE TO CLASS LESSONS IS STRONGLY RECOMMENDED.

Verification of learning
WRITTEN AND ORAL EXAMINATION.
Texts
M. CURZIO, P. LONGOBARDI, M. MAJ - LEZIONI DI ALGEBRA , LIGUORI, 1994, I RISTAMPA 1996.
M. CURZIO, P. LONGOBARDI, M. MAJ - ESERCIZI DI ALGEBRA - UNA RACCOLTA DI PROVE D'ESAME SVOLTE, LIGUORI, NAPOLI, 1995, II EDIZIONE 2011.