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Lecturer(s)
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Dofková Radka, doc. PhDr. Ph.D.
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Zdráhal Tomáš, doc. RNDr. CSc.
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Course content
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Systems of linear equations ( link matrices with systems of linear equations , matrix system and the augmented matrix of the system. Solution of systems of linear equations , systems of equivalent. Theorems for solving systems of linear equations. Matrix homomorphisms of vector spaces . Gauss algorithm , the set of solutions of homogeneous and inhomogeneous systems. ) Polynomials ( Construction structures polynomials ( with coefficients from the field integrity ) . Infer properties of polynomials. Relation between algebraic and functional definition of polynomials over an integral domain . ) The theory of divisibility ( basic concepts of the theory of divisibility in ( commutative ) integral domain (relation " divides " and "being associated " entity within the meaning of divisibility ) . Greatest common divisor and least common multiple , existence and uniqueness . Primes and irreducible elements of Euclidean and Gauss branches integrity. Congruence modulo m Z, areas of congruence classes modulo m , the characteristic of integrity. Fermat's theorem and Euler function and their importance . ) The roots of polynomials ( polynomial root term , Horner , derivative of a polynomial , the multiplicity of the root. Algebraic and transcendental elements over the field . Severability polynomial root factor decomposition product of root factors . Called . Basic algebra theorem ( without proof) . Polynomial in K [ x ] and R [ x] into a product of irreducible elements. )
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Learning activities and teaching methods
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Lecture
- Homework for Teaching
- 50 hours per semester
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Learning outcomes
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Two different approaches (algebraic and functional) to the introduction of polynoms over integrity. Emphasis is placed on active usage of algebraic methods for examining the properties of polynoms.
Ability to abstract relations in numeric fields to more general integral domains.
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Prerequisites
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Successful completion of the course Algebra 1.
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Assessment methods and criteria
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Mark
To understand differences between algebraic and function polynomials, to be able individually to tasks about polynomials, to understand the relationship with other mathematical matter.
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Recommended literature
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BLAŽEK, J. a kol. Algebra a teoretická aritmetika 1. Praha: SPN 1985..
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EMANOVSKÝ, P.: Algebra 2. Olomouc. VUP 2003. ISBN 80-244-0350-7.
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