Algebra
Academic Year 2024/2025 - Teacher: VINCENZO MICALEExpected Learning Outcomes
The purpose of the course is to make students acquire the ability to formalize a problem and to
probe the environment in which to look for any solutions. The course also aims to stimulate the
ability of abstraction and provide the tools to use this abstraction to move from the particular
to the general.
In particular, the course aims to make students acquire the following skills:
Knowledge and understanding: understanding statements and proofs of fundamental theorems of algebra;
develop mathematical skills in reasoning, manipulation and calculation; solve mathematical problems
which, although not common, are of a similar nature to others already known.
Applying knowledge and understanding: to demonstrate algebraic results not identical to those
already known, but clearly related to them; build rigorous proofs; solve algebra problems that
require original thinking; be able to mathematically formalize problems of moderate difficulty,
formulated in natural language, and to take advantage of this formulation to clarify or solve them;
Making judgments: acquiring a conscious autonomy of judgment with reference to the evaluation and
interpretation of the resolution of an algebraic problem; be able to construct and develop logical
arguments with a clear identification of assumptions and conclusions; be able to recognize correct
proofs, and to identify fallacious reasoning.
Communication skills: knowing how to communicate information, ideas, problems, solutions and their
conclusions in a clear and unambiguous way, as well as the knowledge and rationale underlying them;
know how to present scientific materials and arguments, orally or in writing, in a clear and
understandable way.
Learning skills: having developed a greater degree of autonomy in the study.Course Structure
The course consists of lectures, frontal exercises (on the blackboard) and class exercises.
Normally the exercises carried out by the teacher alternate with the theoretical part,
even on the same day. For class exercises, the teacher proposes some exercises to the students
who are invited to solve them by working in small groups; the teacher passes between the desks
helping and suggesting the way to face the exercises. These exercises are essential for
acquiring self-employment and group work skills.
Information for students with disabilities and/or DSA. To guarantee equal opportunities and
in compliance with current laws, interested students can request a personal interview in order
to plan any compensatory and/or dispensatory measures, based on the educational objectives and
specific needs. Students with disabilities and/or DSA they must contact the teacher, the CInAP
contact person of the DMI (Prof. Daniele) and the CInAP sufficiently in advance of the exam
date to communicate that they intend to take the exam taking advantage of the appropriate
compensatory measures (which will be indicated by the CInAP).
Required Prerequisites
Basic knowledge of mathematics present in all high school programs.Attendance of Lessons
Attendance is strongly recommended.
Detailed Course Content
First part (about one third of the course)
a) Elementary set theory.
Sets and operations between sets. Functions. Relations. Equivalence relations. Order relations.
b) Numbers.
Natural numbers.Induction.
Cardinality. Numeralbe sets. |A| < |P(A)|=|2A|. Not numerable sets.
Integers. Greatest common divisor and euclidean algorithm. Bézout identity. Factorization in Z and some consequences. Rational numbers.
Congruence classes. Divisibility criterions. Linear congruences. Euler function and Euler-Fermat theorem.
Real numebres as an ordered field. Complex numbers. Roots of a complex number.
Second part: algebraic structures theory.
a) Ring theory (about one third of the course)
First definitions and examples. Integral domains and fields. Subrings. Homomorphisms. Ideals. Quotients. Homomorphism theorems. Ideal generated by a subset. Prime and maximal ideals. Embedding of a domain in a field and the filed of fractions. Polynomial rings. Polynomial functions and polynomials. Ruffini theorem. Euclidean domains, PID, UFD and relations between these classes. Division between polynomials over a field. Prime and irreducible elements. Bézout identity. GCD and mcm. Gauss lemma and Gauss theorem for A[X], with A UFD. Irreducibility in A[X]. Eisenstein criterion. Irreducibility passing to quotients.
b) Groups theory (about one third of the course)
First definitions and examples. Subgroups. Cyclic groups. Permutations groups. Lagrange theorem. Normal subgroups and quotients. Homomorphisms and related theorems. Cayley's theorem. Action of a group on a set: orbits and stabilizator. Coniugacy classes. Cauchy theorem and Sylow's theorems. Direct sum of groups. Classifications theorem for finite abelian groups.
Contribution of teaching to the objectives of the 2030 Agenda for Sustainable Development In line with the objectives specified on the page
https://asvis.it/goal-e-target-obiettivi-e-traguardi-per-il-2030/ we intend to contribute to Goal 4 objective 4.4 through an informative frontal lesson on the topic.
Textbook Information
2. A. Ragusa - Corso di Algebra (Un approccio amichevole) - Aracne Ed.
3. M. Fontana - S. Gabelli - Insiemi numeri e polinomi - CISU
Learning Assessment
Learning Assessment Procedures
The final exam will consist of a written and an oral test, but it will also take into account what the
student did during the year: - during the year some exercises will be held, in which students will be offered problems to be solved
individually or in small groups and during which the teacher will verify the progress of the test,
suggesting ideas and correcting any errors. It will also be possible to propose tests on the theory
studied. - two tests will then take place, one in itinere and one at the end of the course
(in conjunction with the exams) which, if passed, will give the student exemption from the written
exam. The grade will take into account the written test (or the ongoing tests), the oral learning
and the student's active participation in lessons;
the written tests are considered passed if a grade of not less than 15/30 is obtained.
The final grade does not consist of an average of the marks in the tests, but the oral determines
an increase in the grade of the written test. The vote may also take into account any positive
feedback in the checks carried out during the year.