> I'm working through a book and reached the result > below. Verying the truth > of the statement is claimed to be easy but I've had > little luck. Any ideas > how to proceed? > > Let f(x) be a monic polynomial in Q[x] of degree n > where n happens to be > even. Then there exist polynomials g(x) and h(x) > such that f(x)=g(x)^2-h(x) > and g(x) has degree 1/2*n and h(x) has degree at most > 1/2*n-1. > > Thank you for any comments. > > f is monic with degree 2m As mentioned in a previous post, consider the quotient g when f is divided by x^2 f(x) = x^2 g(x) + L(x) where g is monic with degree 2m-2 and L is linear. By the induction hypothesis, g(x) = h(x)^2 + a x^(m-2) + k(x) where h is monic of degree m-1, a is constant and degree k is < m-2 It is then easy to check that x h(x) + a/2 is the square root of f
We need a recursive algorithm / program to implement this.