Transcendental Numbers by M. Ram Murty & Purusottam Rath

Author:M. Ram Murty & Purusottam Rath
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

Now suppose has class number one. Then by genus theory (see for example, [113]), if − D > 2, then (mod 4) and is necessarily a prime. Moreover as the class number is one, there is only one form (up to equivalence) and which we can take to be

By Dirichlet’s class number formula,

where h 1 is the class number of and ε 1 is the fundamental unit attached to this real quadratic field. The quadratic character χ χ 1 corresponds to the imaginary quadratic field

and we have

where h 2 is the class number of .

We will choose D 1 appropriately. Assuming | D | > D 1 so that (D, D 1) = 1, we obtain from (∗∗),

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