ALGEBRA 1
- Overview
- Assessment methods
- Learning objectives
- Contents
- Full programme
- Bibliography
- Teaching methods
- Contacts/Info
Standard high school mathematics.
Written and oral exam.
The written examination lasts 2 hours and 30 minutes and tipically consists of 4 or 5 exercises divided in subquestions. It is required the ability to establish properties of the algebraic structures, to determine the canonical forms of a square matrix and to justify all the statement given.
The written examination is graded on a scale from 0 to 30. Students need a grade of 14/30 or more to be admitted to the oral examination.
The oral examination usually begins with the discussion of the written test. The student will then be required to present some of the results seen in class. It will be object of evaluation the ability to present a proof in a complete and rigorous.
Passing examination and the final grading depend both on oral and written tests. More detail on course web page.
The aim of the course is to provide the students with a rigorous settlement of well-known arithmetic results and to introduce them to the study of abstract algebraic structures
EXPECTED LEARNING OUTCOMES
At the end of the course the student will be able to:
Use the main algorithms in natural numbers arithmetic and justify them.
Use an adequate mathematical language and expose proofs in a coherent and conscious way
Establish whether a given operation satisfy certain properties
Establish whether a given algebraic structure satisfy certain axioms
Sets
Correspondences. Mappings. Composition of mappings. Associativity of composition. Injective, surjective e bijective mappings. Inverse mapping of a bijective mapping. Relations in a set. Equivalence relations. Partial and total order relations.
Integers.
Operations and their properties. Ordering. Principle of induction. Division. Highest common factor. Bezout identity. Prime numbers characterization. Euclidean algorithm. Prime factorization. Existence of infinite primes. Congruences. Residue classes and operations. The ring of the residue class. Cancellation law and invertible elements modulo n. Linear congruence equations. Chinese remainder theorem. Euler's function.
Groups
Binary operations. Monoids. Groups. Commutative monoids and groups. Residue classes. Plane transformations. Klein's group. Dihedral group. Cancellation law. Multiplication table.
Subgroups. Criterions. Intersection of subgroups. Subgroup generated by a subset.
Action of a group on a set. Trivial action. Transitive actions. Orbits and stabilizers. Orbit equation. Conjugation in a group: conjugacy classes. Centralizer of an element. Center of a finite p-group. Classification of groups of order a square of a prime. Right and left cosets of a subgroup. Index of a subgroup. Lagrange's theorem.
Symmetric group. Cycles. Disjoint cycles. Conjugacy classes in the symmetric group. Transposition. Parity of permutations. Alternating subgroup.
Powers in groups and monoids. Cyclic groups. Order of an element. Subgroups of a cyclic group and their reciprocal position. Number of elements of a given order in a cyclic group.
Normal subgroups. Quotient group. Quotient over the center of a group.
Group and monoid homomorphism. Kernel and image of a homomorphism. Direct and inverse images of subgroups and normal subgroups. Isomorphism theorems. Homomorphisms from cyclic groups. Endomorphisms of cyclic groups.
Product of subgroups. Product of normal subgroups. Internal and external direct product. Direct product of cyclic groups.
Sets
Correspondences. Mappings. Composition of mappings. Associativity of composition. Injective, surjective e bijective mappings. Inverse mapping of a bijective mapping. Relations in a set. Equivalence relations. Partial and total order relations.
Integers.
Operations and their properties. Ordering. Principle of induction. Division. Highest common factor. Bezout identity. Prime numbers characterization. Euclidean algorithm. Prime factorization. Existence of infinite primes. Congruences. Residue classes and operations. The ring of the residue class. Cancellation law and invertible elements modulo n. Linear congruence equations. Chinese remainder theorem. Euler's function.
Groups
Binary operations. Monoids. Groups. Commutative monoids and groups. Residue classes. Plane transformations. Klein's group. Dihedral group. Cancellation law. Multiplication table.
Subgroups. Criterions. Intersection of subgroups. Subgroup generated by a subset.
Action of a group on a set. Trivial action. Transitive actions. Orbits and stabilizers. Orbit equation. Conjugation in a group: conjugacy classes. Centralizer of an element. Center of a finite p-group. Classification of groups of order a square of a prime. Right and left cosets of a subgroup. Index of a subgroup. Lagrange's theorem.
Symmetric group. Cycles. Disjoint cycles. Conjugacy classes in the symmetric group. Transposition. Parity of permutations. Alternating subgroup.
Powers in groups and monoids. Cyclic groups. Order of an element. Subgroups of a cyclic group and their reciprocal position. Number of elements of a given order in a cyclic group.
Normal subgroups. Quotient group. Quotient over the center of a group.
Group and monoid homomorphism. Kernel and image of a homomorphism. Direct and inverse images of subgroups and normal subgroups. Isomorphism theorems. Homomorphisms from cyclic groups. Endomorphisms of cyclic groups.
Product of subgroups. Product of normal subgroups. Internal and external direct product. Direct product of cyclic groups.
P.M. Cohn, Classic Algebra, Wiley.
Exercises, exam problems, and lecture notes available on the class web page.
Frontal lectures. Attending lectures is not mandatory, but strongly recommended.
Lectures are given at the board. Every topic is explained together with exercises useful to understand and apply exposed results. Sometimes the solution is given immediately, sometimes in a subsequent lecture in order to stimulate student to autonomous work. Especially at the end of the course, lectures of summarizing exercises are scheduled, in order to choose the more appropriate method to solve an exercise and to establish connections between different topics.
The exercises presented are often taken from past written examinations: they can be found, together with other selected exercise, on the web page of the course.
For simple and short questions, ask the teacher immediately before or after the class. Email teacher for longer questions.
For further detail go to the web page of the course.