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# Galois Theory of Rigid Correspondences.

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 ,  and .   Next: Purely Iterative Algorithms. Up: Solving the quintic by Previous: Introduction.

Peter Doyle