Let
p
p
be an odd prime and
F
F
a field of characteristic different from
p
p
containing a primitive
p
p
th root of unity. Assume that the Galois group
G
G
of the maximal
p
p
-extension of
F
F
has a finite normal series with abelian factor groups. Then the commutator subgroup of
G
G
is abelian. Moreover,
G
G
has a normal abelian subgroup with pro-cyclic factor group. If, in addition,
F
F
contains a primitive
p
2
{p^2}
th root of unity then
G
G
has generators
{
x
,
y
i
}
i
∈
I
{\{ x,{y_i}\} _{i \in I}}
with relations
y
i
y
j
=
y
j
y
i
{y_i}{y_j} = {y_j}{y_i}
and
x
y
i
x
−
1
=
y
i
q
+
1
x{y_i}{x^{ - 1}} = y_i^{q + 1}
where
q
=
0
q = 0
or
q
=
p
n
q = {p^n}
for some
n
≥
1
n \geq 1
. This is used to calculate the cohomology ring of
G
G
, when
G
G
has finite rank. The field
F
F
is characterized in terms of the behavior of cyclic algebras (of degree
p
p
) over finite
p
p
-extensions.