La Matematica per il 3+2
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This book deals with several topics in algebra useful for computer science applications and the symbolic treatment of algebraic problems, pointing out and discussing their algorithmic nature. The topics covered range from classical results such as the Euclidean algorithm, the Chinese remainder theorem, and polynomial interpolation, to p-adic expansions of rational and algebraic numbers and rational functions, to reach the problem of the polynomial factorisation, especially via Berlekamp’s method, and the discrete Fourier transform. Basic algebra concepts are revised in a form suited for implementation on a computer algebra system.
Groups are a means of classification, via the group action on a set, but also the object of a classification. How many groups of a given type are there, and how can they be described? Hoelder's program for attacking this problem in the case of finite groups is a sort of leitmotiv throughout the text. Infinite groups are also considered, with particular attention to logical and decision problems. Abelian, nilpotent and solvable groups are studied both in the finite and infinite case. Permutation groups and are treated in detail; their relationship with Galois theory is often taken into account. The last two chapters deal with the representation theory of finite group and the cohomology theory of groups; the latter with special emphasis on the extension problem. The sections are followed by exercises; hints to the solution are given, and for most of them a complete solution is provided.