In this section we set up the Galois theory and birational geometry that will be used to describe those field extensions that can be reached by a tower of generally convergent algorithms.
All varieties will be irreducible and complex projective. Let V be a variety, k = K(V) its function field.
An irreducible polynomial p in k[z] determines a
finite field extension 
, where 
is a root of p;
the extension is unique up to isomorphism over k.
To obtain a geometric picture for the field extension,
consider p(z) as a family
of polynomials 
whose coefficients are rational
functions of v. The polynomial p determines a subvariety
which is the closure of the
set of (v,z) such that 
.
The function field 
where
denotes the rational function obtained by projecting
W to 
.
W may be thought of as the graph of a multi-valued function
W(v) which sends v to the roots of 
.
We call such a multi-valued map a rational correspondence.
We say W is a rigid correspondence if its set of
values assumes only one conformal configuration on the Riemann
sphere: i.e. there exists a finite set 
such that the set W(v) is equal to 
for some
Möbius transformation 
depending on v.
In this case we say the field extension is
a rigid extension.
Now let k' denote a finite Galois extension of k with Galois group G.
T 2.1. The field extension k'/k is the splitting field of a rigid extension if and only if there exists:
and
in 
such that
for all g in G.
Proof.
Let k' be the splitting field of a rigid correspondence
.
For simplicity, assume 
is at least 3.
Let 
, i = 1,2,3 denote three
distinct conjugates of under G.
acts triply transitively on the projective line
;
take to be the unique group element
which moves 
to
.
We claim that 
is in
for all g in G. Indeed, is just the
cross-ratio of 
and ,
which is constant by rigidity.
Let 
be the image under of
the conjugates of .
Define 
.
Then 
permutes A, so it is an element
of 
. Because G acts trivially on , 
is a homomorphism; e.g.
and since fixes A pointwise only if g fixes the conjugates of ,
it is faithful; thus we have verified (a-c).
Conversely, given the data (a-c), set 
for any x in with trivial stabilizer in 
; then
is rigid over k and 
.
Cohomological Interpretation.
The map determines an element 
of the Galois cohomology group 
,
which is naturally a subgroup of the Brauer group of k;
condition (c) simply says is the coboundary of ,
so 
.
A geometric formulation of the vanishing of this class is the
following. Let 
denote the rational map of
varieties corresponding to the field extension 
.
Form the Severi-Brauer variety
, where G acts on W by birational
transformations and on via the representation .
Then 
is a flat bundle outside the
branch locus of the map .
We can factor through the inclusion
for any x in
with trivial stabilizer.
The cohomology class of vanishes if and only if
is birational to 
;
in which case 
presents W as a rigid correspondence.
More on Galois cohomology and interpretations of the Brauer group can be found in [6], [7] and [14].