How to obtain generator of principal ideal

Question

Let $f(x) = x^3-x^2+2x+8$, and $K = \mathbb{Q}[x]/(f(x))$.

Using PARI/GP, I knew that the prime $(2)$ is factored into $3$ prime ideals in $K$ as $(2) = P_A P_B P_C$, where

$P_A=(2,x^2/2-x/2)$
$P_B=(2,x^2/2+x/2+3)$
$P_C=(2,x+3)$

According to the website https://www.lmfdb.org/NumberField/3.1.503.1 the class number of $K$ is $1$ so every ideal is principal. So, how to obtain the generator of $P_A, P_B, P_C$? Is there some command available in PARI/GP?

Answer 1

You can do this in Sage via the following code. You can use the online calculator if you don't have Sage on your computer.

R.<x> = PolynomialRing(QQ);
K.<a> = NumberField( x^3 - x^2 + 2*x + 8 );
K.ideal([ 2, x^2/2-x/2 ]).gens_reduced()

Out: (-3/2*a^2 + 7/2*a - 8,)

In fact, the following code will give you the factorization of $(2)$ into principal ideals:

R.<x> = PolynomialRing(QQ);
K.<a> = NumberField( x^3 - x^2 + 2*x + 8 );
P = K.ideal( 2 );
P.factor()

Out: (Fractional ideal (1/2*a^2 - 3/2*a + 3)) * (Fractional ideal (-3/2*a^2 + 7/2*a - 8)) * (Fractional ideal (-2*a^2 + 5*a - 11))