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:

**(a)**- a faithful homomorphism and
**(b)**- an element in such that
**(c)**-
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].