## Thursday, October 01, 2009

### 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=xj, B = xk, C = xm for the variables A,B,C from the sequence x1, x2, x3, ... so that F( xj, xk, xm, ...) ≠ 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 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.

QED

References

rasraster said...

Hi Larry,

This is a really nice and concise proof.

Edwards indicates the general direction the proof should go (induction, starting with A). But since the function in question involves polynomials with mixed terms (e.g., k *(A^2)*(B^3)*(C^4)), I really wasn't sure how to get from (n-1) to n. Your proof is nice and clean and simple and doesn't require imagining how to solve the mixed terms.

Thanks!

rasraster said...

Larry,