Theorem: The product of nonzero polynomials is a nonzero polynomial
(1) This theorem is clearly true in the case of one nonzero polynomial.
(2) Let's assume that it is true up to p-1.
(3) So that the product of p-1 nonzero polynomails is a nonzero polynomial g(x) of degree n so that we have:
g(x) = a0xn + a1xn-1 + ... + an-1x + an
where a0 is nonzero.
(4) Let us assume that f(x) is a nonzero polynomial of degree m so that:
f(x) = b0xm + b1xm-1 + ... + bm-1x + bm
where b0 is nonzero
(5) f(x)*g(x) is nonzero since:
the only term with degree m+n is a0*b0 which cannot be 0.
(6) So, by induction this proposition is true for all products.
- Harold M. Edwards, Galois Theory, Springer, 1984