Proposition

Let $f_1 = y^2 + 2xy - 1$ and $f_2 = x^2 + 1$. Prove that $\ev{f_1, f_2}$ is not a radical ideal.

Solution

$f_1 + f_2 = (x + y)^2$. Thus $(x + y)^2 \in \ev{f_1, f_2}$. We claim that $x + y \notin \ev{f_1, f_2}$.

from sympy import *
from sympy.polys.orderings import monomial_key

x, y = symbols('x y')

print(groebner([y * y + 2 * x * y - 1, x * x + 1], x, y, order='lex'))

gives GroebnerBasis([2*x + y**3 + 3*y, y**4 + 2*y**2 + 1], x, y, domain='ZZ', order='lex').

As the remainder on division of $x + y$ by the Grobner basis is not zero, $x + y \notin \ev{f_1, f_2}$ by Corollary 2 on P.84 (Ideals, Varieties and Algorithms). Therefore, $\ev{f_1, f_2}$ is not radical.