Let K be a field.
Let x1, x2, x3, ... be an infinite sequence of distinct elements of K
Let f(A,B,C,...) be a nonzero polynomial in n variables A,B,C,... with coefficients in K
It is possible to select values A=xj, B = xk, C = xm for the variables A,B,C from the sequence x1, x2, x3, ... so that F( xj, xk, xm, ...) ≠ 0
(1) Assume that f(x) is a nonzero polynomial of one variable with degree m.
(2) Using the Fundamental Theorem of Algebra (see Theorem, here), we know that f(x) has at most m distinct roots.
(3) If we list off m+1 distinct elements of K from the infinite sequence, it is clear that at least one (let us say xr) will not be a root.
(4) So that f(xr) ≠ 0
(5) Assume that this is true up to n-1 variables for F(A,B,C...,Y) so that we know that F(xi, xj, ..., xy) ≠ 0
(6) Let G be a function of n variables so that we have G(A,B,C,...Z)
(7) Let H be a function on the first n-1 variables so that we have H(A,B,C,...Y) = G(A,B,C,...,Y,1)
(8) By assumption, we can find xi, xj, ... xy such that:
H(xi, xj, ..., xy) ≠ 0
(9) But then G(xi, xj, ..., xy, 1) ≠ 0.
- Harold M. Edwards, Galois Theory, Springer, 1984