E. Unique Decomposition of Polynomials into Irreducible Factors.

THEOREM 11. If p(x) is a polynomial in a field F, and if are two factorizations of p(x) into irreducible polynomials each of degree at least one, then r - s and after a suitable change in the order in which the q's are written,

 

Let F (a) be an extension of F in which p i(a) = 0. We may suppose the leading coefficients of the p.( x ) and the q ( x ) to be 1, for by factoring out all leading coefficients and combining, the constant multiplier on each side of the equation must be the leading coefficient of p (x ) and hence can be divided out of both sides of the equation.

 

Since

and

 

F(a) can be 0 only if one of these is 0,

it follows that one of the q^ a), say qt( a), is 0. This gives (see page

 

 

25)

 

 

of two polynomials is 0 only if one of the two is the 0 polynomial, it

Since the product follows that the polynomial within the brackets is 0 so that

p,(x) . . . -pr(x)= q 2( x ) • . qg(x ). If we repeat the above argument

r times we obtain p.(x) = q^x), i = 1, 2, • • > I. Since the remaining

q's must have a product 1, it follows that r - s.