ALGEBRA III

Matematica ALGEBRA III

0512300041
DIPARTIMENTO DI MATEMATICA
EQF6
MATHEMATICS
2022/2023

OBBLIGATORIO
YEAR OF COURSE 2
YEAR OF DIDACTIC SYSTEM 2018
AUTUMN SEMESTER
CFUHOURSACTIVITY
648LESSONS
224EXERCISES
ExamDate
ALGEBRA III19/04/2023 - 14:00
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 CLASS OF ALGEBRA I.
Contents
THE THEORY OF RINGS (24 HOURS):
- 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 (19 HOURS):
- 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 (12 HOURS):
- 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 (18 HOURS):
- 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).
Teaching Methods
THE ALGEBRA III COURSE INCLUDES 64 HOURS OF CLASSROOM TEACHING.
COURSE ATTENDANCE, WHILE NOT MANDATORY, IS STRONGLY RECOMMENDED.
DURING THE LESSONS THEORETICAL TOPICS WILL BE DEALT WITH CONSTANTLY SUPPORTED BY THE PRESENTATION OF
EXAMPLES AND EXERCISES THROUGH WHICH THE METHODS AND CONTEXTS OF USE OF WHAT HAS BEEN EXPLAINED ARE CLARIFIED. FOR THIS REASON, THE EXERCISES ARE INTEGRATED INTO THE SCHEDULED LESSONS.
FINALLY, WITHIN THE 64 HOURS OF TEACHING, SOME LESSONS ARE EXCLUSIVELY DEDICATED TO THE DISCUSSION OF EXERCISES.
Verification of learning
THE AIM OF THE EXAMINATION IS TO EVALUATE THE FAMILIARITY OF THE STUDENT WITH SOME ALGEBRAIC STRUCTURES, LIKE SOME CLASSES OF RINGS, VECTOR SPACES, POLYNOMIALS, FIELDS.
IN THE WRITTEN EXAMINATION THE STUDENT HAS TO SOLVE SOME EXERCISES.
IN THE ORAL EXAMINATION HE HAS TO TALK ABOUT EXAMPLES AND THE PRINCIPAL PROPERTIES OF SOME CLASSES OF RINGS, OF VECTORIAL SPACES, POLYNOMIALS AND FIELDS.
THE FINAL RESULT WILL BE 45% ON THE RESULT OF THE WRITTEN EXAMINATION, 55% ON THE RESULT OF ORAL EXAMINATION.
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
TEACHER'S EMAIL ADDRESS:
PLONGOBARDI@UNISA.IT
Lessons Timetable

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