Determination of the Number of Ideal Classes of a Quadratic Field

David Hilbert

Chapter 19 in The Theory of Algebraic Number Fields, 1998, pp 149-153 from Springer

Abstract: Abstract A remarkable formula for the number h of ideal classes of the quadratic field k results from the expression in Theorem 109 if we evaluate the expression $$\mathop {\lim }\limits_{s = 1} \prod\limits_{\left( p \right)} {\frac{1}{{1 - \left( {\frac{d}{p}} \right){p^{ - s}}}}} $$ on the right hand side in closed form. To this end it is necessary to define the symbol $$\left( {\frac{a}{n}} \right)$$ also in the case where n is a composite positive rational integer.

Date: 1998
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DOI: 10.1007/978-3-662-03545-0_19

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