Matematica | ALGEBRA II

cod. 0512300007

ALGEBRA II

	0512300007
	DIPARTIMENTO DI MATEMATICA

	MATHEMATICS
	2013/2014

CURRICULUM MATEMATICA AD INDIRIZZO APPLICATIVO

	OBBLIGATORIO
	YEAR OF COURSE 2
	YEAR OF DIDACTIC SYSTEM 2010
	PRIMO SEMESTRE

TEACHING UNITS

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

PROFESSORS

	PATRIZIA LONGOBARDI T Curriculum
	MARIA TOTA Curriculum

	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.

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.

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.