Abstract Arithmetic (KMD/E-AG3)

NOTICE: The validity of the information on this page has expired.


  • 6 credits
  • Lecturer: Ing. Tomáš Barot, Ph.D.
  • Lessons (Lectures + Exercises + Seminars): 1 + 0 + 1 [hours/week]
  • Semester: Winter / Summer
  • Language of instruction: English
  • Language of consultation: English
  • Level of qualification (Bc., Mgr.): Bc., Mgr.
  • Method of completion: Examination

Synopsis / description / annotation:

The course introduces methods of construction of number fields - construction of natural numbers using Pean's axioms, natural numbers in the sense of an algebraic structure, a splitting group and a splitting field, construction of the ring of integers and rational numbers, embedding of a lattice into a complete lattice, MacNeil's hull, construction of real numbers in the sense of completion rational numbers, two method of construction - Dedeknid's cuts, topological completion using Cauchy sequences, an algebraic complete field, construction of the field of complex numbers, an algebraic complete field of complex numbers, divisibility in integral domains, special types of integral domains with special arithmetic properties, Gauss rings, a principal ideal ring, Euclidean rings.

Requirements on student:

An active participation in the full-time form of the education, an active approach to solving tasks, the successful passing of the written test of the final exam.

Content:

  1. Construction of natural numbers using Pean’s axioms, natural numbers in the sense of an algebraic structure.
  2. A splitting group and a splitting field.
  3. Construction of the ring of integers and rational numbers.
  4. Embedding of a lattice into a complete lattice, MacNeil’s hull.
  5. Construction of real numbers in the sense of completion rational numbers.
  6. Two method of construction – Dedeknid’s cuts, topological completion using Cauchy sequences.
  7. Algebraic complete field, construction of the field of complex numbers, an algebraic complete field of complex numbers.
  8. Divisibility in integral domains.
  9. Special types of integral domains with special arithmetic properties.
  10. Gauss rings, a principal ideal ring, Euclidean rings.

Time requirements:

  • Being present in classes - 26 h.
  • Self-tutoring - 30 h.
  • Consultation of work with the teacher/tutor (incl. electronic) - 5 h.
  • Preparation for an exam - 30 h.

Prerequisites:

The secondary school knowledge of mathematics is assumed.

Course results:

After finishing the course, in which an understanding of topics of the abstract arithmetic is offered, students will have knowledge about the advanced aspects of these parts of mathematics. Student will have abilities to solve problems using a software support for mathematics.

Assessment methods:

  • Written examination
  • Continuous analysis of student’s achievements

Teaching methods:

  • Computer-based tutoring
  • Dialogic (discussion, dialogue, brainstorming)
  • Individual tutoring

Literature:

  • Lang, S. Undergraduate Algebra. Springer, 2005. ISBN 0-387-22025-9.

Updated: 03. 10. 2022