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:
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 ,  and .