Conguence Of Ankeny-Artin-Chowla Type Of Quartic Fields Q((a+b（-1）^1/2)^1/2)

Let p≡1(mod4), ε=u+v(?)p is the fundamental unit of real quadratic Q((?)p).Ankeny,Artin and Chowla got a beautiful formula about class number h of Q((?)p): h·v/u≡Bp-1/2(mod p)(0.2) Where Bn is the n-th Bernoulli number. subfield.Let f be the conductor of the Galois closure of K4. ε0is the generator of of positive relative units group of K4/K2. h4, h2denote the class number of K4and K2respectively.In order to get Ankeny-Artin-Chowla,we translate the analytic class number into the form ε∩h4/h2=εelliiptic.Where εelliiptic is the ellitic unit.Then we take Kumm-mer’logarithmic derivatives of both sides.Take a prime number π in K2which is prime Where Gkx is the generalized Hurwitz number,(?)’ is prime ideal of Q over (π).