Titlebook: Algebraic Number Theory; Serge Lang Textbook 1994Latest edition Springer Science+Business Media New York 1994 algebraic number theory.anal

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https://doi.org/10.1007/978-3-658-40724-7Using the integrals expressing the zeta function, one can give certain estimates concerning its residue in order to derive asymptotic results relating the class number, regulator, and discriminant of a number field, and notably the following.

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Algebraic IntegersThis chapter describes the basic aspects of the ring of algebraic integers in a number field (always assumed to be of finite degree over the rational numbers .). This includes the general prime ideal structure.

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CompletionsThis chapter introduces the completions of number fields under the p-adic topologies, and also the completions obtained by embedding the number field into the real or complex numbers.

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The Different and DiscriminantThe study of the different and discriminant provides some information on ramified primes, and also gives a sort of duality which plays a role both in the algebraic study of ramification and the later chapters on analytic duality. It also gives a good method for computing the ring of algebraic integers in a number field, as in Proposition 10.

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ParallelotopesThis chapter gives quantitative results concerning the distribution of elements of a number field in parallelotopes.

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