Unit ALGEBRA I
- Course
- Mathematics
- Study-unit Code
- GP006034
- Curriculum
- In all curricula
- Teacher
- Massimo Giulietti
- Teachers
-
- Massimo Giulietti
- Hours
- 47 ore - Massimo Giulietti
- CFU
- 6
- Course Regulation
- Coorte 2026
- Offered
- 2026/27
- Learning activities
- Caratterizzante
- Area
- Formazione matematica teorica
- Sector
- MATH-02/A
- Type of study-unit
- Obbligatorio (Required)
- Type of learning activities
- Attività formativa monodisciplinare
- Language of instruction
- Italian
- Contents
- Classical numerical sets: integers; rationals; reals; complex numbers. Prime numbers. Proofs by induction. Proofs ab absurdo.
Finite and infinite sets: properties and operations. Relations. Applications. Permutations. Cardinality of a set. Countable sets.
Combinatorial calculus. The ring of residue classes modulo an integer n. The Chinese Remainder Theorem. Basics on groups. Actions of groups - Reference texts
- Dikran N. Dikranjan e Maria Silvia Lucido, Aritmetica e algebra. Liguori Editore.
- Educational objectives
- The main goal of the teaching is to provide students with basic knowledge in the field of set theory and some algebraic structures, in order to then be able to undertake subsequent studies. Particular attention is given to the understanding of the arguments and to the rigor in the presentation of concepts and argumentations. Knowledge and comprehension: Mathematical comprehension of the proposed topics, knowledge of the theory developed on sets, functions, cardinality, congruences and fundamental examples discussed on these topics. Methods for verifying knowledge: Written and oral exam. Capacity: Being able to independently read and understand basic Algebra texts, connect arguments, find examples and counterexamples. Being able to produce simple rigorous proofs of mathematical results and problem solving of simple problems related to what has been illustrated in class. Methods for verifying skills: Written and oral exam. Autonomy of judgment: The display of the contents and arguments will be carried out in order to improve the ability of the student to recognize rigorous proofs Communication skills: The presentation of the topics will be organized to allow the acquisition of a good ability to communicate problems, ideas and solutions concerning Algebra, both in written and oral form.
- Prerequisites
- Elementary algebra of the first two years in high school.
- Teaching methods
- Frontal lectures. All theoretical results will be rigorously proved and many related exercises will be proposed.
- Other information
- Learning verification modality
- The exam consists of a written test and an oral test. The written test verifies the ability to produce rigorous demonstrations of problems and statements related to the topics of the course. The oral exam verifies the ability to clearly and rigorously explain some of the course contents. All written tests last about two hours and consist in solving some problems that can also be small parts of theory and serve to check the level of understanding of the topics covered and the ability to connect them. The written test of each session contains 4 problems, one on equivalence relations, one on posets, one on congruences, and one on complex numbers and/or induction principle. The oral exam, lasting about 30 minutes, tends to confirm the level of understanding of the topics covered and of critical study and personal re-elaboration. For information on support services for students with disabilities and / or SLD, visit the page http://www.unipg.it/disabilita-e-dsa
- Extended program
- Classic numeric sets: $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, and $\mathbb{R}$.
Proof by contradiction and proof by induction. The square root of a prime number is an irrational number.
The set $\mathbb{C}$ of complex numbers. Definition of sum and product. Complex conjugate numbers.
Reciprocal of a complex number. Cartesian and trigonometric representation of complex numbers.
Modulus and argument of a complex number. De Moivre's formula. Calculation of the $n$-th roots of unity in the field of complex numbers.
Fundamental Theorem of Algebra (without proof).
Elementary set operations. Cartesian product. The power set of a set.
A set $X$ with $n$ elements has $2^n$ subsets (proof by induction and proof using the characteristic function of $X$: there is a bijection between $\mathcal{P}(X)$ and $\{0,1\}^n$).
Binomial coefficients and their meaning. Pascal's triangle (Tartaglia's triangle).
Mappings (Functions). Injective, surjective, and bijective mappings.
Relations. Order relations. Equivalence relations. Quotient set.
The cardinality of the power set of a set $X$ is strictly greater than the cardinality of $X$.
Prime numbers. Euclidean division. Euclidean algorithm for determining the greatest common divisor between two integers. Bézout's identity.
Euclid's Lemma: if a prime $p$ divides the product of two integers, then $p$ divides at least one of the two integers.
Fundamental Theorem of Arithmetic. Euclid's theorem on the infinitude of the set of prime numbers.
Congruences in $\mathbb{Z}$. Elementary properties. Linear congruential equations. Notes on Diophantine equations.
Systems of congruential equations. Chinese Remainder Theorem.
Proof of the divisibility criteria for 3, 4, 9, 11.
Fermat's Little Theorem.
Euler's totient function $\phi$. Calculation of $\phi(n)$ for every positive integer $n$.
Euler's Theorem.
Algebraic structures with one or more operations. Semigroups. Monoids. Groups.
Examples of abelian and non-abelian groups. The group of invertible matrices with elements in $\mathbb{R}$.
The symmetric group $S_n$ of degree $n$.
The Boolean group of the power set of a set $X$ with respect to the symmetric difference operation.
Subgroups of a group. Criterion to determine if a subset $S$ of a group $G$ is a subgroup of $G$.
Order (or period) $o(x)$ of an element $x$ of a group $G$.
The subgroup generated by $x$. If $o(x)=n$, then $x^h$ has order $n/\gcd(n,h)$.
For any element $x$ of a multiplicative group $G$ of order $n$, $x^n=1$.
Right cosets and left cosets. Lagrange's Theorem: if $H$ is a subgroup of a finite group $G$, then the order of $H$ is a divisor of the order of $G$.
Group actions. Sylow Theorems. - Obiettivi Agenda 2030 per lo sviluppo sostenibile
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