**Defining G(E,F).**
Let *E* be a polynomial equation with *n* variables. Let
*E* be at most quadratic in any variable. Let *F* be a
field or a ring. Define a graph *G*(*E*,*F*) to hava
a vertex set made up of *n*-tuples from *F*^{n}
which are solutions to *E*.
Let the edges in *G* be when two solutions differ in exaclty one
variable.

**Examples of multivariable quadratics:**

2(a^{2} + b^{2} + c^{2} + d^{2})
= (a + b + c + d)^{2}
[Soddy]

a^{2} + b^{2} + c^{2} = 3 a b c
[Markoff]

a_{1}^{2} + a_{2}^{2} +
a_{3}^{2} + ... + a_{n}^{2}
= B a_{1} a_{2} a_{3} ... a_{n}
[Hurwitz]

a^{2} + b^{2} + 2 c^{2} = 4 a b c
[Markoff Brother #1]

a^{2} + 2 b^{2} + 3 c^{2} = 6 a b c
[Markoff Brother #2]

Note that The Soddy Equation was proved by Descartes.

If M is the markoff equation, then G(M,Z) has 5 components. Martin says that for all of the p's he has tried, G(M,Z/p) is connected.

G(M,GF(p^k)) does not seem to be connected for k > 1.

Code: galois.mw and galois.mpl