00005 - Algebra

Learning outcomes

getting used to mathematical reasoning, and to some of the fundamental objects of mathematics: integers and polynomials.



learning the first notions of abstract algebra.

Course contents

the minimum principle, and the principle of induction. examples.

binomial coefficients, and their combinatorial interpretation. binomial formula.

divisibility among integers. prime numbers. greatest common divisor. euclidean algorithm. decomposition in prima factors.



the diofantine equations ax + by = c.



congruences, and their properties. divisibility criteria.



equivalence relations. congruence classes. operations on congruence classes. characterization of invertible classes.




binary operations. commutative rings. calculation rules. examples.




cancellation law. integral domains. invertible elements in a commutative ring. fields. examples.




the field of complex numbers. conjugate and norm of a complex number, and their properties. congruence classes in r modulo a real number. measure of angles. complex numbers in trigonometric form. theorem of de moivre. roots of 1. roots of a complex number.



polynomial with coefficients in a field. addition and multiplication of polynomials. roots of a polynomial.




divisibility among polynomials. division lemma. greatest common divisor of two polynomials. euclidean algorithm.




irreducible polynomial. factorization of a polynomial in irreducible factors.




polynomials of degree 2.




multiplicity of a root of a polynomial. on the number of root of a polynomial. the function associated with a polynomial, and its unicity for infinite fields.



algebraically closed fields. main theorem of algebra.




real polynomials, and their roots.




rational polynomials. algorithms for finding rational roots. reduction modulo a prime, with applications. gauss lemma. eisenstein criterion.




groups. examples.




permutations. the symmetric group on n symbols. cycles. orbits of a permutation. decomposition into disjoint cycles. the sign of a permutation..




subgroups. the theorem of lagrange..




the subgroup generated by one element. cyclic groups. order of an element in a group. the function phi of euler. euler's theorem. the chinese remainder theorem. the fundamental formula on the phi function.

Readings/Bibliography

a.vistoli: Note di algebra. bologna, 1993-94
S. Franciosi, F. de Giovanni, Elementi di Algebra, Aracne, 1992
i.n.herstein: algebra. editori riuniti, roma 1994
e. bedocchi: esercizi di algebra. pitagora editrice bologna, 1995-96.

ulteriore materiale per la preparazione della prova scritta si pu� trovare in tutti gli eserciziari di algebra consultabili in biblioteca, in particolare:
a.alzati - m.bianchi: esercizi di algebra per scienze dell informazione. citt� studi, milano 1991.
a.facchini: sussidiario di algebra e matematica discreta decibel - zanichelli, bologna 1992
m.fontana - s.gabelli: esercizi di algebra aracne editrice, roma, 1993
s.franciosi - f.de giovanni: esercizi di algebra. aracne editrice, roma 1993.
r. procesi ciampi-r.rota: algebra moderna. esercizi. editoriale veschi. masson, milano 1992.
a.rugusa - c.sparacino: esercizi di algebra. zanichelli editore, bologna 1992.

Teaching methods

Lectures and exercise sessions.

Assessment methods

Written and oral exam.

Teaching tools

During the course the students will be handed additional excercise sheets.

Links to further information

http://www.dm.unibo.it/~verardi

Office hours

See the website of Libero Verardi