Mathematics Ii
A.Y. 2026/2027
Learning objectives
The aim of the course is to introduce students to the language and the first notions of discrete mathematics and algebra. In more detail, the course tackles the elementary theory of sets, relations, and functions. It hints at the concept of abstract algebraic structure focusing on the examples of monoids, groups, and rings. It discusses the basic properties of the ring of integers, as well as those of the fields of rational, real and complex numbers. Special attention is paid to the solution of linear congruences and its algorithmic aspects. The course then introduces vector spaces together with linear transformations and their representation by matrices. Finally, the theory is applied to the solution of systems of linear equations, emphasising once again the algorithmic aspects involved.
Expected learning outcomes
Upon course completion, students are expected to understand the basic mathematical formalism of sets, relations, and functions. They are also expected to have an initial degree of familiarity with the concept of abstract algebraic structure. They are expected to understand the basic properties of the ring of integers and of the fields of rational, real, and complex numbers. They are expected to recognise and manipulate vector spaces and their linear transformations. Finally, they are expected to be able to operate with matrices, associating them to systems of linear equations in order to discuss their solvability.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Group 1
Responsible
Course syllabus
1) Algebraic Tools and Basic Algorithms
Integers: Principle of Induction; Integer Division and the Euclidean Algorithm for Calculating the Greatest Common Divisor; Prime Numbers and Prime Factorization; Numbering in Base N.
Polynomials with Real Coefficients and Polynomial Operations; Roots and Their Multiplicities; Irreducible Polynomials; Factorization.
Systems of M Linear Equations in N Unknowns. Resolution with the Gauss-Jordan Escalator Method.
Matrices and Matrix Operations.
2) Abstract Algebra
Sets. Relations between sets and their composition: Equivalence and Order Relations; Applications. Congruence Relations. Modular Arithmetic. Operations between sets. Algebraic Structures, Their Substructures, and Homomorphisms: Groups, Rings (especially Fields and Rings of Polynomials).
3) Linear Algebra
Vector Spaces. Bases. Linear applications and matrices; rank of a matrix. Determinant of a square matrix and its properties. Inverse of a square matrix: existence and computation. Cramer and Rouché-Capelli theorems.
Eigenvalues and eigenvectors, diagonalizability.
Integers: Principle of Induction; Integer Division and the Euclidean Algorithm for Calculating the Greatest Common Divisor; Prime Numbers and Prime Factorization; Numbering in Base N.
Polynomials with Real Coefficients and Polynomial Operations; Roots and Their Multiplicities; Irreducible Polynomials; Factorization.
Systems of M Linear Equations in N Unknowns. Resolution with the Gauss-Jordan Escalator Method.
Matrices and Matrix Operations.
2) Abstract Algebra
Sets. Relations between sets and their composition: Equivalence and Order Relations; Applications. Congruence Relations. Modular Arithmetic. Operations between sets. Algebraic Structures, Their Substructures, and Homomorphisms: Groups, Rings (especially Fields and Rings of Polynomials).
3) Linear Algebra
Vector Spaces. Bases. Linear applications and matrices; rank of a matrix. Determinant of a square matrix and its properties. Inverse of a square matrix: existence and computation. Cramer and Rouché-Capelli theorems.
Eigenvalues and eigenvectors, diagonalizability.
Prerequisites for admission
Basic math skills, such as solving equations and polynomial algebra.
Teaching methods
Theory lectures and classroom exercises.
Tutoring.
Attendance at theory lectures and exercises is strongly recommended.
Tutoring.
Attendance at theory lectures and exercises is strongly recommended.
Teaching Resources
For the Mathematics II course, the following textbook is sufficient:
Discrete Mathematics by Costantino Delizia, Patrizia Longobardi, Mercede Maj, Chiara Nicotera, McGraw Hill, 2009
Other Book (which is enriched with online materials: exercises and various in-depth studies)
What is a Number? An Introduction to Algebra by L. Barbieri Viale, Raffaello Cortina Editore, New Edition, 2026
Discrete Mathematics by Costantino Delizia, Patrizia Longobardi, Mercede Maj, Chiara Nicotera, McGraw Hill, 2009
Other Book (which is enriched with online materials: exercises and various in-depth studies)
What is a Number? An Introduction to Algebra by L. Barbieri Viale, Raffaello Cortina Editore, New Edition, 2026
Assessment methods and Criteria
The final exam consists of a written test.
The written exam will include exercises designed to test the student's ability to solve mathematical problems related to the course content, along with multiple-choice or true/false questions, and open-ended questions requiring proofs of results presented during the course. Instructors will clearly indicate during class which of the proofs presented will be included in the exam.
The duration of each written exam depends on the number, structure, and difficulty of the exercises and questions assigned, but will typically last two and a half hours. During the semester, a two-hour midterm exam will also be scheduled. Passing the exam will entitle the student to fewer exercises in the written exams of the first two available sessions. The results of the written and midterm exams will be communicated to the SIFA through the UNIMIA portal. The final exam grade will be expressed out of thirtieths.
The written exam will include exercises designed to test the student's ability to solve mathematical problems related to the course content, along with multiple-choice or true/false questions, and open-ended questions requiring proofs of results presented during the course. Instructors will clearly indicate during class which of the proofs presented will be included in the exam.
The duration of each written exam depends on the number, structure, and difficulty of the exercises and questions assigned, but will typically last two and a half hours. During the semester, a two-hour midterm exam will also be scheduled. Passing the exam will entitle the student to fewer exercises in the written exams of the first two available sessions. The results of the written and midterm exams will be communicated to the SIFA through the UNIMIA portal. The final exam grade will be expressed out of thirtieths.
MATH-02/B - Geometry - University credits: 5
MATH-03/A - Mathematical Analysis - University credits: 4
MATH-03/A - Mathematical Analysis - University credits: 4
Exercises: 36 hours
Lessons: 48 hours
Lessons: 48 hours
Professors:
Barbieri Viale Luca, Svaldi Roberto
Group 2
Responsible
Lesson period
Second semester
Course syllabus
1) Basic algebraic tools and algorithms
Integers: induction; division; Euclidean algorithm; prime numbers; integer factorization.
Polynomials with real coefficients. Roots. Irreducible polynomials. Factorization of polynomials.
Linear systems; Gauss-Jordan method.
Matrices and their algebra.
2) An outline of abstract algebra
Sets and relations: equivalence relations, partial orderings,maps. Congruence: the integers mod n.
Algebraic structures and their homomorphisms: groups, rings (fields and polynomial rings).
3) Linear Algebra
Vector spaces. Bases. Linear maps and matrices; the rank of a matrix. Determinants of square matrices. Inverse matrices: existence and computing. Cramer and Rouché-Capelli theorems. Eigenvalues and eigenvectors.
Integers: induction; division; Euclidean algorithm; prime numbers; integer factorization.
Polynomials with real coefficients. Roots. Irreducible polynomials. Factorization of polynomials.
Linear systems; Gauss-Jordan method.
Matrices and their algebra.
2) An outline of abstract algebra
Sets and relations: equivalence relations, partial orderings,maps. Congruence: the integers mod n.
Algebraic structures and their homomorphisms: groups, rings (fields and polynomial rings).
3) Linear Algebra
Vector spaces. Bases. Linear maps and matrices; the rank of a matrix. Determinants of square matrices. Inverse matrices: existence and computing. Cramer and Rouché-Capelli theorems. Eigenvalues and eigenvectors.
Prerequisites for admission
Basic knowledge of mathematics, like solving equations and polynomial algebra.
Teaching methods
Frontal lectures about theory and classes of exercises.
Tutoring .
Attendance to theory and exercises classes is strongly recommended
Tutoring .
Attendance to theory and exercises classes is strongly recommended
Teaching Resources
Mainly: Ariel web page and notes of the course.
Suggested book: C. Delizia, P. Longobardi, M. Maj, C. Nicotera - Matematica Discreta - McGraw Hill (2009)
Suggested book: C. Delizia, P. Longobardi, M. Maj, C. Nicotera - Matematica Discreta - McGraw Hill (2009)
Assessment methods and Criteria
The final examination consists of a written test.
The written test will be comprised of exercises designed to test the ability to solve mathematical problems pertaining to the course syllabus, along with multiple-choice or true/false questions, and open-ended questions that will require the student to illustrate the proof of one of the theorems disccused in the course. The teachers will clearly indicate during the lectures which of the proofs presented in the course are examinable.
The duration of each written exam is commensurate with the number, structure, and difficulty of the exercises and questions assigned, but indicatively the exam is expected to last two and a half hours. During the semester in which the course is taught, there will also be a two-hour midterm which, if passed, will entitle the student to take fewer exercises in the written tests of the first two available appeals. The outcomes of the written and midterm exams will be communicated on SIFA through the UNIMIA portal. The final exam grade will be expressed on a scale from 0 to 30 with integer increments; 18 is the minimum passing grade.
The written test will be comprised of exercises designed to test the ability to solve mathematical problems pertaining to the course syllabus, along with multiple-choice or true/false questions, and open-ended questions that will require the student to illustrate the proof of one of the theorems disccused in the course. The teachers will clearly indicate during the lectures which of the proofs presented in the course are examinable.
The duration of each written exam is commensurate with the number, structure, and difficulty of the exercises and questions assigned, but indicatively the exam is expected to last two and a half hours. During the semester in which the course is taught, there will also be a two-hour midterm which, if passed, will entitle the student to take fewer exercises in the written tests of the first two available appeals. The outcomes of the written and midterm exams will be communicated on SIFA through the UNIMIA portal. The final exam grade will be expressed on a scale from 0 to 30 with integer increments; 18 is the minimum passing grade.
MATH-02/B - Geometry - University credits: 5
MATH-03/A - Mathematical Analysis - University credits: 4
MATH-03/A - Mathematical Analysis - University credits: 4
Exercises: 36 hours
Lessons: 48 hours
Lessons: 48 hours
Professors:
Gubbiotti Giorgio, Mazza Carlo
Professor(s)
Reception:
Email contact (usually for Tuesday h. 2-4 p.m.)
Office 2092 - Math Department
Reception:
Tuesday 2.30PM-4.30PM
Diparimento di Matematica "Federigo Enriques" Room 1040
Reception:
By e-mail appointment only.
Office 2103 (second floor) - Dipartimento di Matematica
Reception:
By appointment (to be agreed upon via email)
Room 2102, Dipartimento di Matematica "F. Enriques", Via Saldini 50