Mathematics

Activity type

Formazione Teorica

SSD

MAT/02

Year

2nd year

Semester

2nd semester

CFU

6

Hours distribution

40 classroom hours, 12 training hours

Lecturers

DANIELE VALERI
DANIELE VALERI

Activity type

Formazione Matematica di base

SSD

MAT/02

Year

2nd year

Semester

1st semester

CFU

6

Hours distribution

32 classroom hours, 24 training hours

Lecturers

DANIELE VALERI
DANIELE VALERI

General objectives: to acquire basic knowledge of Algebra.

Specific objectives:

Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results related to:
1) Modular arithmetic.
2) Group Theory.
3) Ring Theory.
4) Field Theory and their extensions.

Apply knowledge and understanding: at the end of the course the student will be able to autonomously handle the initial techniques of abstract algebra and to solve simple problems in the context of the acquired concepts.

Critical and judgmental skills: the student will have the basis to analyze the similarities and relationships with concepts acquired in the first year courses with particular reference to topics concerning linear algebra and the resolution of algebraic equations in the real and complex field.

Communication skills: The learner will have the ability to communicate rigorously the ideas and contents shown in the course.

Learning skills: the acquired knowledge will allow a study, individual or given in a subsequent course in order to acquire more advanced concepts related to the main algebraic structures.

Module: arithmetic and Groups
General objectives: acquire the basic knowledge of Algebra
related to topics of ring theory and field theory.


Specific objectives:


Knowledge and understanding: at the end of the course the student will have
acquired the basic notions and results related to:

1) Theory of Rings.

2) Field theory and their extensions.


Applying knowledge and understanding: at the end of the course the student
will be able to autonomously handle the initial techniques
of the theory of rings and fields, and to solve simple ones
problems relating to rings of polynomials, Euclidean rings, finite extensions



Critical and judgmental skills: the student will have the basis for
analyze the similarities and relationships with notions acquired in the courses
of the first year with particular reference to topics concerning
linear algebra and the resolution of algebraic equations in the fields of
real and complex numbers.


Communication skills: The learner will have the ability to communicate
rigorously the ideas and contents presented in the course.

Learning skills: the knowledge gained will allow one
to study, individually or in subsequent courses, more
advanced topics related to the main algebraic structures.


Module: rings and fields
General objectives: acquire the basic knowledge of Algebra
related to topics of elementary number theory and group theory.


Specific objectives:


Knowledge and understanding: at the end of the course the student will have
acquired the basic notions and results related to:

1) Modular arithmetic.

2) Group theory.


Applying knowledge and understanding: at the end of the module the student
will be able to autonomously handle the initial techniques
of group theory and to solve simple problems of
modular arithmetic


Critical and judgmental skills: the student will have the basis for
analyze the similarities and relationships with notions acquired in the courses
of the first year with particular reference to topics concerning
linear algebra and transformation groups.


Communication skills: The learner will have the ability to communicate
rigorously the ideas and contents presented in the course.

Learning skills: the knowledge gained will allow one
to study, individually or in subsequent courses, more
advanced topics related to the main algebraic structures.

Module: arithmetic and Groups
It useful for the student to know and master the basics imparted in the first courses of linear algebra.


Module: rings and fields
It useful for the student to know and master the basics imparted in the first courses of mathematical analysis and linear algebra.

Module: arithmetic and Groups
Third Part. Elements of ring theory

Rings, ideals, quotients, homomorphisms, theorems of homomorphism and isomorphism, field of fractions of a domain. - Euclidean domains, principal ideal domains, unique factorization domains, Gauss integers, whole sum of two squares, prime ideals and maximal ideals; divisibility in domains; prime and irreducible elements.

Rings of polynomials: universal property, polynomials with coefficients in a domain, the Euclidean property of monic polynomials, quotient rings of polynomial rings, Gaussian lemma, Eisenstein criterion and other irreducibility criteria; the irreducible elements of Z[x], unique factorization in Z[x].

Fourth part. Elements of field theory

Extensions of fields, algebraic and transcendental elements, finite and algebraic extensions, degree of extension, splitting field of a polynomial.

Algebraically closed fields, the fundamental theorem of algebra, multiple roots and derivative criterion, classification of finite fields, Frobenius morphism, n-th roots of unity and cyclotomic extensions.

Constructions with row and compass (the problems of the trisection of the angle, the quadrature of the circle, the rectification of the circumference and the duplication of the cube).

A brief introduction to Galois theory in 0 characteristic.


Module: rings and fields
First part. Arithmetic on Z and modular

Euclidean division, MCD, Euclidean algorithm and Bezout identity, prime numbers, fundamental theorem of arithmetic.

Congruences, invertible elements of Z / mZ, Euler's function, Euler-Fermat's theorem, Fermat's Small Theorem, equations and systems of congruential equations, Chinese remainder theorem, RSA.

Second part. Elements of group theory

Groups, subgroups and normal subgroups, quotients, homomorphisms, homomorphism and isomorphism theorems, conjugated elements, Lagrange and Cayley theorems.

Cyclic groups and their subgroups, dihedral groups, symmetric groups (writing of a permutation in disjoint cycles, even and odd class, conjugated permutations in the symmetric group). Direct and semi-direct product of groups. Finite p-groups.

Finitely generated Abelian groups and their classification.
Group actions on a set. Sylow theorems and applications.

Module: arithmetic and Groups
Israel Herstein, Algebra, Editori Riuniti
Michael Artin, Algebra, Bollati Boringhieri
S. Weintraub, Galois Theory
Lecture Notes on Groups Theory by J. Milne


Module: rings and fields
Israel Herstein, Algebra, Editori Riuniti
Michael Artin, Algebra, Bollati Boringhieri
S. Weintraub, Galois Theory
Lecture Notes on Groups Theory by J. Milne

Module: arithmetic and Groups
N/D
Module: rings and fields
N/D

Module: arithmetic and Groups
The course consists of lectures and exercise sessions.


Module: rings and fields
The course consists of lectures and exercise sessions.

Module: arithmetic and Groups
Attending is not mandatory.


Module: rings and fields
Attending is not mandatory.

Module: arithmetic and Groups
The exam aims to evaluate learning through a written test (consisting in solving problems of the same type as those carried out in the exercises) and an oral test (consisting in the discussion of the most relevant topics illustrated in the course).

The written test will last about three hours and can be replaced by two intermediate tests, both lasting at least two hours, the first of which will take place in the middle and the second at the end of the course. The first intermediate test will focus mainly on the topics of arithmetic and group theory, the second on the remaining topics of the course.

Students who have obtained a grade of not less than 17/30 in the written test (or the average of the two intermediate tests) are admitted to the oral exam. The minimum grade to pass the exam is 18/30. The student must demonstrate that he has acquired sufficient knowledge of the topics of all parts of the program.

To achieve a score of 30/30 cum laude, the student must demonstrate that he has acquired excellent knowledge of all the topics covered during the course and be able to link them in a logical and coherent manner.


Module: rings and fields
The exam aims to evaluate learning through a written test (consisting in solving problems of the same type as those carried out in the exercises) and an oral test (consisting in the discussion of the most relevant topics illustrated in the course).

The written test will last about three hours and can be replaced by two intermediate tests, both lasting at least two hours, the first of which will take place in the middle and the second at the end of the course. The first intermediate test will focus mainly on the topics of arithmetic and group theory, the second on the remaining topics of the course.

Students who have obtained a grade of not less than 18/30 in the written test (or the average of the two intermediate tests) are admitted to the oral exam. The minimum grade to pass the exam is 18/30. The student must demonstrate that he has acquired sufficient knowledge of the topics of all parts of the program.

To achieve a score of 30/30 cum laude, the student must demonstrate that he has acquired excellent knowledge of all the topics covered during the course and be able to link them in a logical and coherent manner.

Module: arithmetic and Groups
See the webpage of the course 2023/2024: https://www1.mat.uniroma1.it/people/desole/didattica/2023-algebra1/index.html


Module: rings and fields
See the webpage of the course 2023/2024: https://www1.mat.uniroma1.it/people/desole/didattica/2023-algebra1/index.html

Module: arithmetic and Groups
N/D
Module: rings and fields
N/D