Algebra

Most of the topics are covered in the undergraduate course MATE 4008, Introduction to Algebraic structures and the graduate course MATE 6201, Abstract Algebra.

Topics:

• Integers: basic properties, induction, the Euclidian algorithm, the Fundamental Theorem of Arithmetic.
• Rational numbers: construction of rational numbers from integers, relation between integers and rational numbers.
• Real numbers: brief introduction of their properties and their relation with the rational numbers.
• Complex numbers: construction of complex numbers using the set of real numbers. Fundamental Theorem of Algebra (knowledge of the proof is not required at this level).
• Polynomials: division of polynomials, divisibility properties regarding coefficient type: integers, rationals, reals or complex, characterization of irreducible polynomials of real or complex coefficients, roots of polynomials with integer coefficients. Solution of quadratic, cubic and quartic equations.
• Groups: definition and basic properties, subgroups, cyclic groups, group of permutations, direct products, group homomorphism, normal groups, quotient group and Sylow theorems.
• Rings: Definition and basic properties, ring of polynomials, domains, ring homomorphisms, maximal and prime ideals, quotient rings, prime and irreducible elements, commutative and Euclidean rings, principal ideals rings, unique factorization.
• Fields: finite fields, algebraic and transcendental extensions of a field, degree of an algebraic extension. Field extensions using the roots of a polynomial, decomposition fields.

Bibliography:

• A survey of Modern Algebra, Birkhoff-McLane Macmillan International Pub.
• Abstract Algebra, D.S. Dummit and R.M. Foote, Wiley 3rd Edition.
• Basic Algebra, N. Jacobson, W. H. Freeman publisher.
• Topics in Algebra, Herstein , John Wiley & Sons.
• Abstract Algebra, Herstein, Second edition, Macmillan International Pub.
• Algebra, T. Hungerford, Springer-Verlag.
Previous exams:

Spring-2005