Classification of some Galois fields with a fixed Polya index

The Polya group P o ( K ) of a Galois number field K coincides with the subgroup of the ideal class group C l ( K ) of K consisting of all strongly ambiguous ideal classes. We prove that there are only finitely many imaginary abelian number fields K whose "Polya index" [ C l ( K ) : P o ( K ) ] is a fixed integer. Accordingly, under GRH, we completely classify all imaginary quadratic fields with the Polya indices 1 and 2. Also, we unconditionally classify all imaginary biquadratic and imaginary tri-quadratic fields with the Polya index 1. In another direction, we classify all real quadratic fields K of extended R-D type (with possibly only one more field K ) for which P o ( K ) = C l ( K ) . Our result generalizes Kazuhiro's classification of all real quadratic fields of narrow R-D type whose narrow genus numbers are equal to their narrow class numbers.

This is a joint work with Amir Akbary (University of Lethbridge).