  Separations among the first order logic ${\cal R}ing(0,+,*)$ of finite
residue class rings, its extensions with generalized quantifiers, and in the
presence of a built-in order are shown, using algebraic methods from class
field theory. These methods include classification of spectra of sentences over
finite residue classes as systems of congruences, and the study of their
$h$-densities over the set of all prime numbers, for various functions $h$ on
the natural numbers. Over ordered structures the logic of finite residue class
rings and extensions are known to capture DLOGTIME-uniform circuit complexity
classes ranging from $AC^0$ to $TC^0$. Separating these circuit complexity
classes is directly related to classifying the $h$-density of spectra of
sentences in the corresponding logics of finite residue classes. We further
give general conditions under which a logic over the finite residue class rings
has a sentence whose spectrum has no $h$-density. One application of this
result is that in ${\cal R}ing(0,+,*,<) + M$, the logic of finite residue class
rings with built-in order and extended with the majority quantifier $M$, there
are sentences whose spectrum have no exponential density.
