Algebra for Computer Science

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.