A proof is given of an explicit
Dirichlet-type class number formula for imaginary cyclic quartic fields obtained in
1980 by Hudson and Williams and, in a slightly different form, by Setzer. The
Hudson-Williams formula is used to study the solvability of the quaternary quadratic
form
16pk
= x2+ 2qu2+ 2qv2+ qw2,
xw
= av2− 2buv − au2, (x,u,v,w,p) = 1
for exponents k ≥ 1. Included is a table from which every class number h(k) of the
quartic field k = Q(i), q ≡ 5 (mod8) a prime, may be determined
for q < 10000. Finally, a quartic analog of the well-known result that the
number of quadratic residues in (0,p∕2) exceeds the number in (p∕2,p) if
p ≡ 3 (mod4) is proven using one of Dirichlet’s less well-known class number
formulas.
), q ≡ 5 (mod 8) a prime, may be determined
for q < 10000. Finally, a quartic analog of the well-known result that the
number of quadratic residues in (0,p∕2) exceeds the number in (p∕2,p) if
p ≡ 3 (mod 4) is proven using one of Dirichlet’s less well-known class number
formulas.
