Algebra 1977/78, 1978/79, 1986/87, 1987/88, 1988/89, 1989/90, 1996/97. For 1st year mathematics students. Official name: Algebra. Program 1996/97.
PRELIMINARIES The Greek alphabet Sets Axioms of Peano and the induction principle Some logics Set operations with an infinite number of operands The cartesian product Binomial numbers THE INTEGERS The euclidean algorithm Continued fractions Fundamental relations for continued fractions Linear diophantine equations in two variables Prime numbers RELATIONS AND FUNCTIONS Functions Relations Equivalence relations GROUP THEORY Groupoids and semigroups Groups Rings and modules Congruences in a semigroup Isomorphisms and homomorphisms Congruences in a group and normal subgroups All cyclic groups Theorems of Lagrange and Fermat Theorems of isomorphy Direct products of groups The Chinese remainder theorem Number theoretic congruences COMMUTATIVE RINGS Polynomial rings Symmetric polynomials Divisibility in a commutative ring Congruences in a commutative ring Prime ideals and maximal ideals Factorial rings If A is integer, every finite subgroup of A* is cyclic Roots of unity The quotient field of an integral domain FIELD THEORY The minimal subfield of a field and the characteristic Algebraic and transcendent elements Adjunction of roots Algebraic extensions Existence of an algebraic closure Separable extensions Simple algebraic extensions Normal extensions The fundamental theorem of Galois theory Extension of homomorphisms Finite fields The field with 4 elements The field with 8 elements