Teachings
60/64/170  ALGEBRA 1
Academic Year 2019/2020
Free text for the University
 Professor

STEFANO MONTALDO (Tit.)
 Period

First Semester
 Teaching style

Convenzionale
 Lingua Insegnamento

ITALIANO
Informazioni aggiuntive
Course  Curriculum  CFU  Length(h) 

[60/64] MATHEMATICS  [64/00  Ord. 2017] PERCORSO COMUNE  8  64 
Objectives
Algebra 1 should be considered the first course for a student enrolled in the Mathematics Degree both for the content and for a certain math mental form that the student acquires during his study.
The course concerns elementary mathematics, almost apparently already known by students from middle schools, but treated from a point of view and depth different from those to which high school students are accustomed. During the course, both in classroom and in self study, the student learns to use mathematical formalism correctly to describe the elementary objects and constructions of Mathematics. In particular, the student should acquire the following knowledge and skills.
KNOWLEDGE AND UNDERSTANDING
Know the language of set theory. Know and understand equivalence classes and quotient sets.
Know natural, integer, rational, real, and complex numbers. Know the rudiments of the number theory. Have a good knowledge of polynomials in one and more variables.
APPLYING KNOWLEDGE AND UNDERSTANDING
Ability to properly formulate mathematical statements and construct rigorous simple demonstrations. Know how to work with equivalence classes and quotient sets. Know how to use the properties of operations defined in numeric sets correctly. Know and know how to use the arithmetic of integers. Know how to manipulate polynomials in one and more variables.
MAKING JUDGEMENTS
Knowing how to recognize when a logical procedure is correct.
COMMUNICATION SKILLS
Learn how to use the modern language of algebra to properly communicate scientific results.
LEARNING SKILLS
Ability to learn how to solve independently complex exercises. Ability to be able to read and understand a text that uses the language of mathematics.
Prerequisites
To follow the course you do not need specific knowledge of Mathematics, the notions learned in any school are more than enough. Nonetheless some students state that their preliminary knowledge is not enough. Actually, it is not the knowledge which is missing but a certain habit of mathematical reasoning. Many students come to the university with the idea that math is a great formulary from which one must find the formula that best suits the resolution of the proposed exercise. They remain baffled when they realize that during the Algebra 1 lessons there is no new formula to learn, exercise resolution consists in knowing how to use logic and intuition to check if a given statement is true or false. So the most important prerequisite is to have a healthy desire to understand why a procedure works and never assume anything for granted. With this attitude you usually get good results.
Contents
Detailed plan:
http://people.unica.it/montaldo/files/2008/06/programmaalgebra11718.pdf
SETS, RELATIONS AND FUNCTIONS.
The concept of set
Operations between sets
Relations and functions
Composition of applications
The natural numbers and the induction principle
Finite and infinite sets
Equivalence relations
Partitions and binomial coefficients
Relations of order and preorder
Axiom of choice
Cartesian products
Cardinal numbers
INTEGERS, RATIONAL, REAL AND COMPLEX
Integers
The rational numbers
The real numbers
Complex numbers
THE INTEGER ARITHMETIC
Prime numbers
Greatest common divisor and least common multiple
The Euclidean division
The fundamental theorem of arithmetic
Congruence in Z
Congruenzal equations and Diophantine equations
Some criteria of divisibility
Fermat's theorem
Euler function and Euler's theorem
The Fermat numbers and Mersenne numbers
Perfect numbers and amicable numbers
Distribution of prime numbers
Sums of two squares
POLYNOMIALS
Definition of the ring of polynomials
Polynomial division
Roots of a polynomial
Factoring polynomials
Irreducible polynomials
Homogeneous polynomials
Teaching Methods
Blackboard during frontal lessons. For the student's preparation at home, the teacher has a website dedicated to students (people.unica.it/montaldo) where they can find teacher notes, the lesson register, weekly exercises to be done at home and corrected in class during the exercises. Trying to solve at home the exercises is one of the most important steps for learning the math method. The student should not make the mistake of following the exercises without having tried the exercises (for several hours). The individual study, led by the teacher, plays a key role in the first year of Mathematics. Students are encouraged to discuss each other either to compare solutions found in exercises or to see if the topics taught in the lesson have been properly understood and assimilated. The course includes 32 frontal lectures of 2 hours each and 16 exercises of 2 hours each. The latter are assisted by an expert tutor.
Verification of learning
The goal of the exam is to check the level of achievement of the abovementioned training objectives.
Examination consists of a written test and an oral exam.
The written test (3 hours) consists in a selection of 6 exercises similar to the homework assigned during the course. Each exercise is worth 5 points and the written test is deemed passed if the final score is not less than 18. In this test mainly occur: knowledge and understanding; knowledge and understanding applied; the ability to learn.
The oral exam (usually 40 minutes) consists of a discussion of at least three arguments. The student must demonstrate that they have understood and assimilated the arguments (knowledge and understanding, knowledge and understanding applied). In addition, students must demonstrate that they are able to explain the concepts and demonstrations learned during the course (Communication Skills). To this end it is necessary that the student, during the oral exam, write on the chalkboard and explain all the steps that it is following to reach the conclusion of an argument (independent judgment). It is advisable to answer exactly the questions without rambling and writing immediately all the necessary details (independent judgment, communication skills).
The final grade after the oral interview is assigned according to the following Docimological Table.
Insufficient: the student proves he did not understand many of the basic constructions of the discipline.
1824: the student knows almost all the arguments submitted during the examination; demonstrates that has understood and assimilated the arguments sufficiently.
2528: the student knows all the arguments submitted during the examination; demonstrates that has understood and assimilated well the arguments; the student uses mathematical language correctly.
2930: the student knows very well all the arguments submitted during the examination; demonstrates that has understood and assimilated very well the arguments; the student uses mathematical language correctly; is able to explain the concepts learned.
30 e lode: the student knows perfectly all the arguments submitted during the examination; demonstrates that has understood and assimilated in depth all the topics; the student uses mathematical language correctly; is able to explain the concepts learned.
Texts
Textbook
Dikran Dikranjan, Maria Silvia Lucido, Aritmetica e algebra, Liguori editore
Other books
C. Casolo, Dispense del corso di ALGEBRA I, http://web.math.unifi.it/users/casolo/dispense/algebra1_11.pdf
L. Cerlienco, Numeri e poco altro, http://corsi.unica.it/matematica/files/2014/10/algebra11.pdf
More Information
The teacher receives the students, when present in the office, every day of the week both in the morning and in the afternoon, even without appointment. If students wishes to do so, they can request an appointment by email: montaldo@unica.it
Webpage
people.unica.it/montaldo
Our University provides support for students with specific learning disability (SLD). Those interested can find more informations at this link:
http://corsi.unica.it/matematica/infodsa/