Thursday, October 01, 2009

Products of Nonzero Polynomials

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

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.



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