# Rule Of Signs

This is my 100th post. I decided to write a math post for my 100th
. Originally, I was going to write about p-adic fields, Hasse prin
ciple, localization of commutative rings, local vs global, &c. Or
about Frobenius morphism and Galois groups. But soon I realized th
at it would be too long for those cases. And I don't want to write
long, so... I'm going to write a simple, but interesting one. It's
called Descartes' rule of signs.

------------------------------------------------------------------
Thm. The number of positive roots is at most the number of sign ch
anges in the polynomial's coefficients.
------------------------------------------------------------------

DISCLAIMER: Discard zero coefficients.

For negative roots, use p(-x). (maximum)
For nonreal roots, use n-(positives)-(negatives). (minimum)

Because of the conjugacy, the number of nonreal roots is always ev
en. So if the number of sign changes is zero or one, the number of
positive roots equals the number of sign changes.

The reason is this. Let the maximum number of positive roots be n.
Then the possibilities are n, n-2, n-4, ... because of the possibi
lities of nonreal roots. So if n is either 0 or 1, that is the exa
ct number of positive roots.


## Example

For p(x) = x^3 + x^2 - x - 1, ++-- so it has exactly one positive
root. (+ "+-" -)

p(-x) = -x^3 + x^2 + x - 1, -++- so it has two or zero negative ro
ots. ("-+" "+-")

So the possibilities are [3 real (+--) roots] or [1 real (+) and 2
nonreal roots].

For q(x) = x^3 - 1, +- so it has exactly one positive root.

q(-x) = -x^3 - 1, -- so it has exactly zero negative root.

So the exact number of nonreal roots of q(x) is 3-1-0=2. You would
already know that the roots are 1, w, w^2. These form the finite f
ield F_3.