1 Number theory -- 1.1 Divisibility -- 1.2 Congruences -- 1.3 The theorems of Fermat, Euler and Wilson -- 1.4 Squares and the quadratic reciprocity theorem -- 1.5 The Gaussian integers -- 1.6 Algebraic numbers -- 1.7 Appendix. Primitive elements and a theorem by Gauss -- Literature -- 2 Number theory and computing -- 2.1 The cost of arithmetic operations -- 2.2 Primes and factoring -- 2.3 Pseudo-random numbers -- Literature -- 3 Abstract algebra and modules -- 3.1 The four operations of arithmetic -- 3.2 Modules -- 3.3 Module morphisms. Kernels and images -- 3.4 The structure of finite modules -- 3.5 Appendix. Finitely generated modules -- Literature -- 4 The finite Fourier transform -- 4.1 Characters of modules -- 4.2 The finite Fourier transform -- 4.3 The finite Fourier transform and the quadratic reciprocity law -- 4.4 The fast Fourier transform -- Literature -- 5 Rings and fields -- 5.1 Definitions and simple examples -- 5.2 Modules over a ring. Ideals and morphisms -- 5.3 Abstract linear algebra -- Literature -- 6 Algebraic complexity theory -- 6.1 Polynomial rings in several variables -- 6.2 Complexity with respect to multiplication -- 6.3 Appendix. The fast Fourier transform is optimal -- Literature -- 7 Polynomial rings, algebraic fields, finite fields -- 7.1 Divisibility in a polynomial ring -- 7.2 Algebraic numbers and algebraic fields -- 7.3 Finite fields -- Literature -- 8 Shift registers and coding -- 8.1 The theory of shift registers -- 8.2 Generalities about coding -- 8.3 Cyclic codes -- 8.4 The BCH codes and the Reed-Solomon codes -- 8.5 Restrictions for error-correcting codes -- Literature -- 9 Groups -- 9.1 General theory -- 9.2 Finite groups -- Literature -- 10 Boolean algebra -- 10.1 Boolean algebras and rings -- 10.2 Finite Boolean algebras -- 10.3 Equivalence classes of switching functions -- Literature -- 11 Monoids, automata, languages -- 11.1 Matrices with elements in a non-commutative algebra -- 11.2 Monoids and languages -- 11.3 Automata and rational languages -- 11.4 Every rational language is accepted by a finite automaton -- Literature -- References.