Reelle biquadratische Zahlkörper mit ungerader Klassenzahl. (Real biquadratic number fields with odd class number) (Q1113943)

Using Kubota's class number relations and Hasse's formula for the number of ambiguous ideal classes, the author obtains explicit sufficient conditions for a real bicyclic biquadratic number field to have odd class number. Let \(p_ i\) (resp., \(q_ i)\) be primes congruent to \(1\quad (resp.\quad 3)\quad (mod\quad 4).\) Let \(K={\mathbb{Q}}(\sqrt{m_ 1},\sqrt{m_ 2})\), where \(m_ 1\) and \(m_ 2\) are square-free positive integers. Then K has odd class number in each of the following \(cases:\) (m\({}_ 1,m_ 2)= \) \((q_ 1,2)\), \((q_ 1,q_ 2)\), \((q_ 1q_ 2,q_ 1q_ 3)\), \((2q_ 1,2q_ 2);\) \((2,p_ 1)\) with \(p_ 1\equiv 5(mod 8);\) \((p_ 1,p_ 2)\) with \((p_ 1/p_ 2)=-1;\) \((q_ 1,p_ 1)\) with \((q_ 1/p_ 1)=-1\), or with \((q_ 1/p_ 1)=1\) and \(p_ 1\equiv 5(mod 8);\) \((2,q_ 1q_ 2)\) with \(q_ 1q_ 2\equiv 5(mod 8)\), or \(q_ 1\equiv q_ 2\equiv 3(mod 8);\) \((p_ 1,2q_ 1)\) with \((2q_ 1/p_ 1)=-1\), or with \((2q_ 1/p_ 1=1\) and \(p_ 1\equiv 5(mod 8)\) \((p_ 1,q_ 1q_ 2)\) with \((q_ 1q_ 2/p)=-1\), or with \((p_ 1/q_ 1)=(p_ 1/q_ 2)=-1.\) The author also obtains several necessary conditions in case K has odd class number and \((m_ 1,m_ 2)\) does not appear on the above list.