Rings of Invariants
                                     and
                    Inseparable Forms of Algebras
                                     over
                          the Steenrod Algebra

                      ByClarence W. Wilkerson

                             Purdue University


                 Dedicated to the memory of J.F. Adams



   Theconcept of the maximal torus plays a key role in the classifica-
tion of compact Lie groups.  Likewise the role of the cohomology of
the classifying space BTn  of the n-torus in the study of characteristic
classes has been long understood via the splitting theorems.  With the
work of Adams-Wilkerson (1) it was realized that the cohomology rings
H (BT n; Fp) as algebras over the Steenrod algebra Ap   were universal
in a strong technical sense. This feature was presaged by Serre (13),
Quillen (11), and Wilkerson (14), and is greatly extended by later work
of Carlsson, Miller, Lannes, Zarati, Schwartz, and Dwyer-Wilkerson.
   TheAdams-Wilkerson work provides an algebraic analogue of the ex-
istence of a maximal torus, in a suitable category of algebras with an
action  of  the  Steenrod  algebra.  More  precisely, it  provides  for  each
graded Fp- algebra R  which is an integral domain of finite transcen-
dence!degree n over Fp and equipped with an "unstable" action of the
mod!p!Steenrod algebra Ap , an embedding
!
!!                       tR  : R    ! H (BT n;Fp )
!
which!is!an Ap -map of algebras and is an invariant of R   and its Ap  -
action.!Furthermore, H (BTn ;Fp) is algebraic and normal over R , via
the!morphism!tR  .
!  Inthis paper, we denote H (B Tn; Fp) together with its Ap -action as
S[V!],!the symmetric algebra ona ndimensional Fpvector space V;
for!V!concentrated in degree 2 (or degree 1 for p = 2).
!  Theembedding tR   has additional properties,depending on the prop-
erties!of!R  as an algebra over the Steenrod algebra:
!
The author was supported in part bythe National Science Foundation, the Wayne