  A major open problem in current Galois theory is to characterize those
profinite groups which appear as absolute Galois groups of various fields.
Obtaining detailed knowledge of the structure of quotients and subgroup
filtrations of Galois groups of $p$-extensions is an important step toward a
solution. We illustrate several techniques for counting Galois $p$-extensions
of various fields, including pythagorean fields and local fields. An expression
for the number of extensions of a formally real pythagorean field having Galois
group the dihedral group of order 8 is developed. We derive a formula for
computing the $\mathbb{F}_p$-dimension of an $n$-th graded piece of the
Zassenhaus filtration for various finitely generated pro-$p$ groups, including
free pro-$p$ groups, Demushkin groups and their free pro-$p$ products. Several
examples are provided to illustrate the importance of these dimensions in
characterizing pro-$p$ Galois groups. We also show that knowledge of small
quotients of pro-$p$ Galois groups can provide information regarding the form
of relations among the group generators.
