Nonzero Polynomials with Distinct Parameters

The following is taken from Harold M. Edwards in his book Galois Theory.

Theorem:

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

Then:

It is possible to select values A=x_j, B = x_k, C = x_m for the variables A,B,C from the sequence x₁, x₂, x₃, ... so that F( x_j, x_k, x_m, ...) ≠ 0

Proof:

(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 x_r) will not be a root.

(4) So that f(x_r) ≠ 0

(5) Assume that this is true up to n-1 variables for F(A,B,C...,Y) so that we know that F(x_i, x_j, ..., x_y) ≠ 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 x_i, x_j, ... x_y such that:

H(x_i, x_j, ..., x_y) ≠ 0

(9) But then G(x_i, x_j, ..., x_y, 1) ≠ 0.

QED

References

• Harold M. Edwards, Galois Theory, Springer, 1984