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Topic: Simple (?) Polynomial Result
Replies: 10   Last Post: Jan 16, 2011 1:42 PM

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Ara M Jamboulian

Posts: 71
Registered: 12/6/04
Re: Simple (?) Polynomial Result
Posted: Sep 14, 2010 2:34 AM
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> 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.

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