1. SylabUZ
2. Faculty of Mathematics, Computer Science and Econometrics
3. winter term 2019/2020
4. Mathematics - First-cycle studies leading to Bachelor's degree
5. General Algebra

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General Algebra - course description

Aim of the course

In the end of this course the students know and understand the basic theorems concerning groups, rings, fields and lattices theory and they can applicate and use the notions and theorems from the abstract algebra in codes, cryptography and combinatorics.

Prerequisites

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Linear Algebra 1 and 2.

Scope

1. Prime numbers, The Basic Theorem of Arithmetics, congruences of integer numbers, Euler Totient function, Euler theorem. Definitions and properties of operations in the algebraic structures.
2. Groups, abelian groups, cyclic groups, subgroups, permutation groups, torsion and torsion-free groups. Cayley’s theorem and Lagrange’s Thorem for groups. Morphisms of groups, normal subgroups, simple groups, congruences in groups. Quotient groups, Isomorphism theorem for groups. Sylow’s Thorem.
3. Rings, subrings, ideals, congruences in rings, quotient rings. Isomorphism theorem for rings, principal ideals, prime ideals, Maximal ideals. Chinese theorem. Fields, simple fields, finite fields.
4. Polynomial rings in one and many indeterminates, polynomial roots, symmetric polynomials. Bezout’s theorem, Gauss’s theorem, Eisenstein-Shönemann’s criterion. Algebraic elements over a field, minimal polynomial. Extensions of fields. Fields algebraically closed. Hilbert’s zeros Theorem.
5. Lattices, modular and distributive lattices, sublattices, examples. Dedekind-Birkhoff theorem. Boolean algebras.

Teaching methods

Traditional lectures; Solving appropriate selected exercises in the class.

Learning outcomes and methods of theirs verification

Outcome description	Outcome symbols	Methods of verification	The class form

Assignment conditions

Verifying the level of preparation of students and their activities during the classes. The student has to receive the positive grade from two tests with tasks of different difficulty which help to assess whether students have achieved effects of the course in a minimum degree (40% of the final grade). Written exam (60% of the final grade).

Recommended reading

1. Białynicki-Birula, Zarys algebry, BM tom 63, PWN, Warszawa, 1987.
2. M. Bryński, Algebra dla studentów matematyki, PWN, Warszawa 1987.
3. Gleichgewicht, Algebra, Oficyna GiS, 2002.
4. W. J. GILBERT, W. K. NICHOLSON, MODERN ALGEBRA WITH APPLICATIONS, A JOHN WILEY & SONS, INC., PUBLICATION (http://cs.ioc.ee/~margo/aat/Gilbert W.J., Nicholson W.K. Modern algebra with applications (2ed., Wiley, 2004)(ISBN 0471414514)(347s).pdf)
5. J. Rutkowski, Algebra abstrakcyjna w zadaniach, PWN, Warszawa, 2000.

Further reading

1. G.Birkhoff,T.C.Bartee, Współczesna algebra stosowana, PWN,Warszawa,1983.
2. S. Burris, H. P. Sankappanavar , A Course in Universal Algebra, (http://orion.math.iastate.edu/cliff/BurrisSanka.pdf)
3. M. Bryński, J. Jurkiewicz, Zbiór zadań z algebry, PWN, Warszawa 1985.
4. A.I. Kostrykin, Wstęp do algebry, cz. I, III, PWN, Warszawa, 2005.
5. R. Lidl, Algebra dla przyrodników i inżynierów, PWN, Warszawa 1983.
6. A. Mostowski, M. Stark, Algebra wyższa, cz. I, II, III, PWN, 1966.

Notes

Modified by dr Robert Dylewski, prof. UZ (last modification: 19-09-2019 13:42)