  Hilbert's Nullstellensatz characterizes polynomials that vanish on the
vanishing set of an ideal in C[x]. In the free algebra C<X> the vanishing set
of a two-sided ideal I is defined in a dimension-free way using images in
finite-dimensional representations of C<x>/I. In this article
Nullstellens\"atze for a simple but important class of ideals in the free
algebra - called tentatively rationally resolvable here - are presented. An
ideal is rationally resolvable if its defining relations can be eliminated by
expressing some of the X variables using noncommutative rational functions in
the remaining variables. Whether such an ideal I satisfies the Nullstellensatz
is intimately related to embeddability of C<x>/I into (free) skew fields. These
notions are also extended to free algebras with involution. For instance, it is
proved that a polynomial vanishes on all tuples of spherical isometries iff it
is a member of the two-sided ideal I generated by 1-\sum_j X_j^* X_j. This is
then applied to free real algebraic geometry: polynomials positive semidefinite
on spherical isometries are sums of Hermitian squares modulo I. Similar results
are obtained for nc unitary groups.
