Introduction of Transcendental Methods into the Arithmetic of Number Fields
In 1840, Dirichlet, in his pioneering paper “Recherches sur diverses applications de l’analyse infinitésimale à la théorie des nombres” (Crelles Journal, Vol. 19. Werke Vol. 1 p. 411), showed how the powerful methods of the analysis of continuous variables can be used in the solution of purely arithmetic problems. These methods have become of great significance for the arithmetic of number fields. Even today the problem of the class number and the problem of the distribution of prime ideals are still only approachable by these transcendental methods, and at this time they still completely evade a purely arithmetic treatment.
KeywordsPrime Ideal Class Number Dirichlet Series Integral Ideal Residue Class
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