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B005473 - ALGEBRA I
Main information
Teaching Language
Course Content
Suggested readings
Learning Objectives
Prerequisites
Teaching Methods
Type of Assessment
Course program
Academic Year 2018-19
Coorte 2018 - 3-years First Cycle Degree (DM 270/04) in MATHEMATICS
Course year
First year - Annualità singola
Belonging Department
Mathematics and Computer Science "Ulisse Dini" (DIMAI)
Course Type
Single education field course
Scientific Area
MAT/02 - ALGEBRA
Credits
9
Teaching Hours
90
Teaching Term
17/09/2018 ⇒ 14/06/2019
Attendance required
No
Type of Evaluation
Final Grade
Course Content
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Course program
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Lectureship
Teaching Language
Italian
Course Content
Basic aspects of the technical mathematical language, specially from abstract perspective: sets, relations, maps, equivalences.
Review of the fundamentals of arithmetic.
Introduction to the algebraic structure of a ring. Idelas, homomorphisms, quotients, rings of polynomials.
Review of the fundamentals of arithmetic.
Introduction to the algebraic structure of a ring. Idelas, homomorphisms, quotients, rings of polynomials.
Suggested readings (Search our library's catalogue)
Course Notes, available at:
http://web.math.unifi.it/users/casolo/dispense/algebra1_17.pdf
http://web.math.unifi.it/users/casolo/dispense/algebra1_17.pdf
Learning Objectives
The corse aims at furnishing the basic mathematical language, to be used in all the other course, as well as giving the basics about rings, with emphasis on polynomials and applications to arithmetic questions.
Prerequisites
As a first year Course, it does not assume any previous University course.
Teaching Methods
Lectures
Type of Assessment
The exam consists of a written and an oral part.
Course program
Part I: Sets, operations with sets, cartesian product; maps and their special properties, bijections, composition, inverse maps.
Integers: induction principle, b-adic representations, division, GCM and Binet's formula, prime numbers, fundamental theorem of Arithmetic, euclidean algorithm., finite sets and their orders, binomial coefficients, Newton's formula; introduction to complex numbers.
Binary operations, equivalence relations, quotient set, partitions: order relations, max, min, sup and inf.
Cardinality of a set. The set of rationals is enumerable. Cantor's Theorem.
Diofantine equation of first degree, congruences, congruence classes, quotient set, compatibility with the two operations, Fermat's Little Theorem, congruence equations, Chines remainders Theorem.
Part II: Rings and their basic properties, powers and multiples, subrings, zero-divisors. Integral domains, fields, direct products of rings. Ideals, ideals in Z, principal ideals, sum of ideals , ideals and fields. Homomorphisms and isomorphisms of rings, image and kernel.
Important rings: the ring of residue classe, invertible and zero-divisors in Z/nZ, characteristic of a ring, ring of fractions of an integral domain. division and principal ideals, prime and irreducible elements, factorizations, Factorial domains and their characterization. Maximal ideals and prime ideals, the case of a Principal Ideal Domani. Euclidean domain, every Principal Ideal Domain is factorial. Gauss integers, sum of squares.
Polynomial rings, degree, universal property of polynomial rings, formal construction of polynomial and power series rings. Euclidea division of polynomials. A polynomial ring with coefficients in a field is euclidean, Euclid's algorithm for polynomials, factorizations. Ruffini's Theorem, distinct roots, irreducible polynomials. Wilson's Theorem. Primitive polynomials and Gauss Lemma, factorizations in Z[x] and Q[x], Eisenstein's criterion.
Quotient rings: quotients modulo a prime or a maximal ideal. First homomorphism theorem and correspondence theorem, ideal of a quotient ring. Factor rings of a PID, irreducible elements and maximal ideal, examples of finite fields. , esempi di campi finiti. Simple extensions of fields.
Integers: induction principle, b-adic representations, division, GCM and Binet's formula, prime numbers, fundamental theorem of Arithmetic, euclidean algorithm., finite sets and their orders, binomial coefficients, Newton's formula; introduction to complex numbers.
Binary operations, equivalence relations, quotient set, partitions: order relations, max, min, sup and inf.
Cardinality of a set. The set of rationals is enumerable. Cantor's Theorem.
Diofantine equation of first degree, congruences, congruence classes, quotient set, compatibility with the two operations, Fermat's Little Theorem, congruence equations, Chines remainders Theorem.
Part II: Rings and their basic properties, powers and multiples, subrings, zero-divisors. Integral domains, fields, direct products of rings. Ideals, ideals in Z, principal ideals, sum of ideals , ideals and fields. Homomorphisms and isomorphisms of rings, image and kernel.
Important rings: the ring of residue classe, invertible and zero-divisors in Z/nZ, characteristic of a ring, ring of fractions of an integral domain. division and principal ideals, prime and irreducible elements, factorizations, Factorial domains and their characterization. Maximal ideals and prime ideals, the case of a Principal Ideal Domani. Euclidean domain, every Principal Ideal Domain is factorial. Gauss integers, sum of squares.
Polynomial rings, degree, universal property of polynomial rings, formal construction of polynomial and power series rings. Euclidea division of polynomials. A polynomial ring with coefficients in a field is euclidean, Euclid's algorithm for polynomials, factorizations. Ruffini's Theorem, distinct roots, irreducible polynomials. Wilson's Theorem. Primitive polynomials and Gauss Lemma, factorizations in Z[x] and Q[x], Eisenstein's criterion.
Quotient rings: quotients modulo a prime or a maximal ideal. First homomorphism theorem and correspondence theorem, ideal of a quotient ring. Factor rings of a PID, irreducible elements and maximal ideal, examples of finite fields. , esempi di campi finiti. Simple extensions of fields.