Carmine MONETTA | ALGEBRA III

cod. 0512300041

ALGEBRA III

	0512300041
	DEPARTMENT OF MATHEMATICS
	EQF6
	MATHEMATICS
	2024/2025

PERCORSO SHARED STUDY PATHWAY

	OBBLIGATORIO
	YEAR OF COURSE 2
	YEAR OF DIDACTIC SYSTEM 2018
	AUTUMN SEMESTER

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
	CARMINE MONETTA Curriculum

EXAMS

	Exam	Date	Session
	ALGEBRA III	18/02/2025 - 14:00	SESSIONE ORDINARIA
	ALGEBRA III	18/02/2025 - 14:00	SESSIONE DI RECUPERO

	Objectives
	GENERAL PURPOSE: THE AIM OF THE COURSE IS TO PROVIDE FURTHER BASIC NOTIONS ON ALGEBRAIC STRUCTURES. KNOWLEDGE AND UNDERSTANDING: THE PRIMARY OBJECTIVE OF THE COURSE IS TO COMPLETE AN INITIAL KNOWLEDGE OF THE MAIN ALGEBRAIC STRUCTURES, WITH PARTICULAR ATTENTION TO RINGS AND FIELDS, WHILE AT THE SAME TIME ACCUSTOMING THE STUDENT TO FORMULATING PROBLEMS AND REASONING RIGOROUSLY. THE STUDENT WILL BE ASKED: - THE KNOWLEDGE OF FURTHER NOTABLE PROPERTIES RELATING TO RINGS AND VECTOR SPACES; - IN-DEPTH STUDY OF POLYNOMIALS AND FIELDS; - KNOWLEDGE OF THE FIRST ELEMENTS OF GALOIS THEORY. ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING: THE AIM OF THE COURSE IS TO ENABLE THE STUDENT TO: - RECOGNIZE AND USE ALGEBRAIC STRUCTURES SUCH AS RINGS, VECTOR SPACES AND ABOVE ALL FIELDS; - STUDY POLYNOMIALS KNOWING HOW TO IDENTIFY ROOTS AND FACTORS; - HIGHLIGHT PROPERTIES OF FIELD EXTENSIONS; - CONSTRUCT SPLITTING FIELDS OF POLYNOMIALS OF POSITIVE DEGREE; - CORRECTLY AND RIGOROUSLY STATE DEFINITIONS AND THEOREMS REGARDING THE CONTENTS OF THE COURSE ITSELF, AND PRECISELY RECONSTRUCT THE RELATED PROOFS; - SOLVE EXERCISES AND PROBLEMS RELATING TO THE CONTENTS PROPOSED DURING THE COURSE. AUTONOMY OF JUDGEMENT: THE STUDENT WILL BE GUIDED TO LEARN IN A CRITICAL AND RESPONSIBLE MANNER EVERYTHING THAT IS ILLUSTRATED IN THE CLASSROOM AND TO IMPROVE HIS JUDGMENT SKILLS ALSO THROUGH THE STUDY OF THE INDICATED TEACHING MATERIAL. THE STUDENT WILL BE ABLE TO EVALUATE THE CORRECTNESS OF PROPOSITIONS, IDENTIFYING APPROPRIATE EXAMPLES AND DEMONSTRATING THEM WITH RIGOROUS ARGUMENTATION, OR CONFUTING THEM WITH THE PRODUCTION OF COUNTEREXAMPLES. COMMUNICATION SKILLS: THE STUDENT WILL BE ABLE TO USE A FORMAL, CLEAR BUT RIGOROUS MATHEMATICAL LANGUAGE TO DESCRIBE THE CONCEPTS EXAMINED DURING THE COURSE, TO FORMULATE POSSIBLE CONJECTURES, TO IDENTIFY PROOF TECHNIQUES. LEARNING ABILITY: THE STUDENT WILL BE ABLE TO: - UNDERSTAND AND USE FORMAL MATHEMATICAL LANGUAGE; - CONSTRUCT EXAMPLES AND/OR COUNTEREXAMPLES; - SOLVE PROBLEMS RELATING TO THE TOPICS COVERED; - UNDERSTAND, ANALYZE AND RECONSTRUCT THE STRUCTURE OF MATHEMATICAL PROOFS; - USE THE FUNDAMENTAL IDEAS OF SOME DEMONSTRATIONS EVEN IN CONTEXTS OTHER THAN THOSE PROPOSED DURING THE COURSE.

GENERAL PURPOSE:
THE AIM OF THE COURSE IS TO PROVIDE FURTHER BASIC NOTIONS ON ALGEBRAIC STRUCTURES.

KNOWLEDGE AND UNDERSTANDING:
THE PRIMARY OBJECTIVE OF THE COURSE IS TO COMPLETE AN INITIAL KNOWLEDGE OF THE MAIN ALGEBRAIC STRUCTURES, WITH PARTICULAR ATTENTION TO RINGS AND FIELDS, WHILE AT THE SAME TIME ACCUSTOMING THE STUDENT TO FORMULATING PROBLEMS AND REASONING RIGOROUSLY.
THE STUDENT WILL BE ASKED:
- THE KNOWLEDGE OF FURTHER NOTABLE PROPERTIES RELATING TO RINGS AND VECTOR SPACES;
- IN-DEPTH STUDY OF POLYNOMIALS AND FIELDS;
- KNOWLEDGE OF THE FIRST ELEMENTS OF GALOIS THEORY.

ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING:
THE AIM OF THE COURSE IS TO ENABLE THE STUDENT TO:
- RECOGNIZE AND USE ALGEBRAIC STRUCTURES SUCH AS RINGS, VECTOR SPACES AND ABOVE ALL FIELDS;
- STUDY POLYNOMIALS KNOWING HOW TO IDENTIFY ROOTS AND FACTORS;
- HIGHLIGHT PROPERTIES OF FIELD EXTENSIONS;
- CONSTRUCT SPLITTING FIELDS OF POLYNOMIALS OF POSITIVE DEGREE;
- CORRECTLY AND RIGOROUSLY STATE DEFINITIONS AND THEOREMS REGARDING THE CONTENTS OF THE COURSE ITSELF, AND PRECISELY RECONSTRUCT THE RELATED PROOFS;
- SOLVE EXERCISES AND PROBLEMS RELATING TO THE CONTENTS PROPOSED DURING THE COURSE.

AUTONOMY OF JUDGEMENT:
THE STUDENT WILL BE GUIDED TO LEARN IN A CRITICAL AND RESPONSIBLE MANNER EVERYTHING THAT IS ILLUSTRATED IN THE CLASSROOM AND TO IMPROVE HIS JUDGMENT SKILLS ALSO THROUGH THE STUDY OF THE INDICATED TEACHING MATERIAL.
THE STUDENT WILL BE ABLE TO EVALUATE THE CORRECTNESS OF PROPOSITIONS, IDENTIFYING APPROPRIATE EXAMPLES AND DEMONSTRATING THEM WITH RIGOROUS ARGUMENTATION, OR CONFUTING THEM WITH THE PRODUCTION OF COUNTEREXAMPLES.

COMMUNICATION SKILLS:
THE STUDENT WILL BE ABLE TO USE A FORMAL, CLEAR BUT RIGOROUS MATHEMATICAL LANGUAGE TO DESCRIBE THE CONCEPTS EXAMINED DURING THE COURSE, TO FORMULATE POSSIBLE CONJECTURES, TO IDENTIFY PROOF TECHNIQUES.

LEARNING ABILITY:
THE STUDENT WILL BE ABLE TO:
- UNDERSTAND AND USE FORMAL MATHEMATICAL LANGUAGE;
- CONSTRUCT EXAMPLES AND/OR COUNTEREXAMPLES;
- SOLVE PROBLEMS RELATING TO THE TOPICS COVERED;
- UNDERSTAND, ANALYZE AND RECONSTRUCT THE STRUCTURE OF MATHEMATICAL PROOFS;
- USE THE FUNDAMENTAL IDEAS OF SOME DEMONSTRATIONS EVEN IN CONTEXTS OTHER THAN THOSE PROPOSED DURING THE COURSE.

	Prerequisites
	GOOD KNOWLEDGE OF THE TOPICS CONTAINED IN THE ALGEBRA I/ALGEBRA II COURSE.

	Contents
	THE THEORY OF RINGS (21 HOURS = 15 + 6): - NILPOTENT AND IDEMPOTENT ELEMENTS. - THE RING OF INTEGRAL QUATERNIONS, THE DIVISION RING OF REAL QUATERNIONS - MAXIMAL IDEALS. - THE FIELD OF QUOTIENTS OF AN INTEGRAL DOMAIN. - THE ENDOMORPHISM RING OF AN ABELIAN GROUP. - UNIQUE FACTORIZATION DOMAINS, THEIR CHARACTERIZATION. EXAMPLES. EXISTENCE OF GCD AND LCM. - PRINCIPAL DOMAINS. EXAMPLES, PROPERTIES. EVERY PRINCIPAL DOMAIN IS A UNIQUE FACTORIZATION DOMAIN. - EUCLIDEAN DOMAINS. EXAMPLES AND PROPERTIES. EVERY EUCLIDEAN DOMAIN IS A PRINCIPAL DOMAIN. EUCLIDEAN ALGORITHM. POLYNOMIALS (15 HOURS = 11 + 4): - THE RING OF FORMAL SERIES AND OF POLYNOMIALS OVER A RING WITH UNIT. THE UNIVERSAL PROPERTY. GAUSS’ LEMMA. THE EISENSTEIN CRITERIUM. THE FUNDAMENTAL POLYNOMIAL OF A FINITE FIELD. ROOTS OF A POLYNOMIAL. VECTOR SPACES (13 HOURS = 8 + 5): - ISOMORPHIC VECTOR SPACES. - DECOMPOSITIONS. - FINITE DIMENSIONAL VECTOR SPACES. EXISTENCE OF VECTOR SPACES OF ARBITRARY DIMENSION. - THE ADDITIVE STRUCTURE OF A VECTOR SPACE AND OF A DIVISION RING. THE THEORY OF FIELDS (20 HOURS = 12 + 8): - 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. GENERALITIES ON GALOIS THEORY (3 HOURS = 2 + 1).

Contents

THE THEORY OF RINGS (21 HOURS = 15 + 6):
- NILPOTENT AND IDEMPOTENT ELEMENTS.
- THE RING OF INTEGRAL QUATERNIONS, THE DIVISION RING OF REAL QUATERNIONS
- MAXIMAL IDEALS.
- THE FIELD OF QUOTIENTS OF AN INTEGRAL DOMAIN.
- THE ENDOMORPHISM RING OF AN ABELIAN GROUP.
- UNIQUE FACTORIZATION DOMAINS, THEIR CHARACTERIZATION. EXAMPLES. EXISTENCE OF GCD AND LCM.
- PRINCIPAL DOMAINS. EXAMPLES, PROPERTIES. EVERY PRINCIPAL DOMAIN IS A UNIQUE FACTORIZATION DOMAIN.
- EUCLIDEAN DOMAINS. EXAMPLES AND PROPERTIES. EVERY EUCLIDEAN DOMAIN IS A PRINCIPAL DOMAIN. EUCLIDEAN ALGORITHM.

POLYNOMIALS (15 HOURS = 11 + 4):
- THE RING OF FORMAL SERIES AND OF POLYNOMIALS OVER A RING WITH UNIT. THE UNIVERSAL PROPERTY. GAUSS’ LEMMA. THE EISENSTEIN CRITERIUM. THE FUNDAMENTAL POLYNOMIAL OF A FINITE FIELD. ROOTS OF A POLYNOMIAL.

VECTOR SPACES (13 HOURS = 8 + 5):
- ISOMORPHIC VECTOR SPACES.
- DECOMPOSITIONS.
- FINITE DIMENSIONAL VECTOR SPACES. EXISTENCE OF VECTOR SPACES OF ARBITRARY DIMENSION.
- THE ADDITIVE STRUCTURE OF A VECTOR SPACE AND OF A DIVISION RING.

THE THEORY OF FIELDS (20 HOURS = 12 + 8):
- 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.

GENERALITIES ON GALOIS THEORY (3 HOURS = 2 + 1).

	Teaching Methods
	THE ALGEBRA III COURSE INCLUDES 72 HOURS OF CLASSROOM TEACHING, 48 OF LESSONS AND 24 OF EXERCISES. ATTENDING THE COURSE, ALTHOUGH NOT MANDATORY, IS STRONGLY RECOMMENDED. DURING THE LESSONS THEORETICAL ISSUES WILL BE ADDRESSED ALONG WITH THE PRESENTATION OF EXAMPLES AND EXERCISES THROUGH WHICH THE METHODS AND CONTEXTS OF USE OF WHAT IS EXPLAINED ARE CLARIFIED. FOR THIS REASON THE EXERCISES ARE INTEGRATED INTO THE PLANNED LESSONS. FINALLY, WITHIN THE 72 HOURS OF TEACHING, SOME LESSONS EXCLUSIVELY DEDICATED TO CARRYING OUT EXERCISES ARE PROVIDED.

Teaching Methods

THE ALGEBRA III COURSE INCLUDES 72 HOURS OF CLASSROOM TEACHING, 48 OF LESSONS AND 24 OF EXERCISES.
ATTENDING THE COURSE, ALTHOUGH NOT MANDATORY, IS STRONGLY RECOMMENDED.
DURING THE LESSONS THEORETICAL ISSUES WILL BE ADDRESSED ALONG WITH THE PRESENTATION OF
EXAMPLES AND EXERCISES THROUGH WHICH THE METHODS AND CONTEXTS OF USE OF WHAT IS EXPLAINED ARE CLARIFIED. FOR THIS REASON THE EXERCISES ARE INTEGRATED INTO THE PLANNED LESSONS.
FINALLY, WITHIN THE 72 HOURS OF TEACHING, SOME LESSONS EXCLUSIVELY DEDICATED TO CARRYING OUT EXERCISES ARE PROVIDED.

	Verification of learning
	THE EXAM TEST IS AIMED AT ASSESSING AS A WHOLE THE KNOWLEDGE AND ABILITY TO UNDERSTAND THE CONCEPTS PRESENTED IN LESSONS, AS WELL AS THE ABILITY TO APPLY SUCH KNOWLEDGE IN THE STUDY OF ALGEBRAIC STRUCTURES, SUCH AS PARTICULAR CLASSES OF RINGS, VECTOR SPACES, POLYNOMIALS AND FIELDS . THE EXAM CONSISTS OF A SELECTIVE WRITTEN TEST AND AN ORAL INTERVIEW. THE WRITTEN TEST INCLUDES SOME EXERCISES. WITH THE ORAL INTERVIEW, THE KNOWLEDGE ACQUIRED REGARDING THE ALGEBRAIC STRUCTURES STUDIED WILL BE ASSESSED. IN THE FINAL ASSESSMENT, EXPRESSED IN THIRTIETHS, THE EVALUATION OF THE WRITTEN TEST WILL WEIGH FOR 45% WHILE THE ORAL INTERVIEW FOR THE REMAINING 55%. PRAISE MAY BE AWARDED TO STUDENTS WHO DEMONSTRATE THEY ARE ABLE TO INDEPENDENTLY APPLY KNOWLEDGE AND SKILLS ACQUIRED EVEN IN CONTEXTS DIFFERENT THAN THOSE PROPOSED IN LESSONS.

Verification of learning

THE EXAM TEST IS AIMED AT ASSESSING AS A WHOLE THE KNOWLEDGE AND ABILITY TO UNDERSTAND THE CONCEPTS PRESENTED IN LESSONS, AS WELL AS THE ABILITY TO APPLY SUCH KNOWLEDGE IN THE STUDY OF ALGEBRAIC STRUCTURES, SUCH AS PARTICULAR CLASSES OF RINGS, VECTOR SPACES, POLYNOMIALS AND FIELDS .
THE EXAM CONSISTS OF A SELECTIVE WRITTEN TEST AND AN ORAL INTERVIEW. THE WRITTEN TEST INCLUDES SOME EXERCISES. WITH THE ORAL INTERVIEW, THE KNOWLEDGE ACQUIRED REGARDING THE ALGEBRAIC STRUCTURES STUDIED WILL BE ASSESSED.
IN THE FINAL ASSESSMENT, EXPRESSED IN THIRTIETHS, THE EVALUATION OF THE WRITTEN TEST WILL WEIGH FOR 45% WHILE THE ORAL INTERVIEW FOR THE REMAINING 55%. PRAISE MAY BE AWARDED TO STUDENTS WHO DEMONSTRATE THEY ARE ABLE TO INDEPENDENTLY APPLY KNOWLEDGE AND SKILLS ACQUIRED EVEN IN CONTEXTS DIFFERENT THAN THOSE PROPOSED IN LESSONS.

	Texts
	M. CURZIO, P. LONGOBARDI, M. MAJ - LEZIONI DI ALGEBRA , LIGUORI, 1994, I REPRINT 1996, II EDITION 2014. M. CURZIO, P. LONGOBARDI, M. MAJ - ESERCIZI DI ALGEBRA - UNA RACCOLTA DI PROVE D'ESAME SVOLTE, LIGUORI, NAPOLI, 1995, II EDITION 2011.

	More Information
	TEACHERS' EMAIL ADDRESS: PLONGOBARDI@UNISA.IT CMONETTA@UNISA.IT WEBSITE: HTTPS://DOCENTI.UNISA.IT/004793 HTTPS://DOCENTI.UNISA.IT/028874

Lessons Timetable

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Carmine MONETTA | ALGEBRA III