UTM Ideals Varieties and Algorithm - Chapter 1 Section 5 Exercise 15

Problem:

Solution:

Part (a) is obvious

$\begin{eqnarray*} & & \frac{f}{GCD(f, f')} \\ &=& \frac{c(x - a_1)^{r_1} \cdots (x - a_n)^{r_n}}{(x - a_1)^{r_1 - 1} \cdots (x - a_n)^{r_n - 1}} \\ &=& c(x - a_1) \cdots (x - a_n) \\ &=& f_{\text{red}} \end{eqnarray*}$

Part (b) is just an application of the formula.

print "Problem 15"
f = polynomial.from_string("x^11 - x^10 + 2x^8 - 4x^7 + 3x^5 - 3x^4 + x^3 + 3x^2 - x - 1")
fd = f.derivative()
g = polynomial.polynomial_gcd(f, fd)
(q, r) = polynomial.polynomial_divide(f, g)
print q

The square free part of the polynomial is $x^5 + x^2 - x - 1$