Example 8.1.6 (Dedekind)
Let
![$ K=\mathbf{Q}(a)$](img502.png)
be the cubic field defined by a root
![$ a$](img163.png)
of the polynomial
![$ f = x^3 + x^2 - 2x+8$](img569.png)
. We will use
, which implements the algorithm
described in the previous section, to show that
![$ 2$](img25.png)
is an essential
discriminant divisor for
![$ K$](img9.png)
.
> K<a> := NumberField(x^3 + x^2 - 2*x + 8);
> OK := MaximalOrder(K);
> Factorization(2*OK);
[
<Prime Ideal of OK
Basis:
[2 0 0]
[0 1 0]
[0 0 1], 1>,
<Prime Ideal of OK
Basis:
[1 0 1]
[0 1 0]
[0 0 2], 1>,
<Prime Ideal of OK
Basis:
[1 0 1]
[0 1 1]
[0 0 2], 1>
]
Thus
![$ 2\O _K=\mathfrak{p}_1\mathfrak{p}_2\mathfrak{p}_3$](img570.png)
, with the
![$ \mathfrak{p}_i$](img408.png)
distinct.
Moreover, one can check that
![$ \O _K/\mathfrak{p}_i\cong \mathbf{F}_2$](img571.png)
. If
![$ \O _K=\mathbf{Z}[a]$](img492.png)
for some
![$ a\in\O _K$](img494.png)
with minimal polynomial
![$ g$](img135.png)
, then
![$ \overline{g}(x)\in\mathbf{F}_2[x]$](img572.png)
must be a product of three
distinct
linear factors, which is impossible.