Canonical defining polynomial for number fields (reviewed)

Every number field $K$ can be represented as $K = \Q[X]/P(x)$ for some monic $P\in\Z[X]$, called a defining polynomial for $K$. Among all such defining polynomials, we define the reduced defining polynomial as follows.

Recall that for a monic polynomial $P(x) = \prod_i(x-\alpha_i)$, the $T_2$ norm of $P$ is $T_2(P) = \sum_i |\alpha_i|^2$.

• Let $L_0$ be the list of (monic integral) defining polynomials for $K$ that are minimal with respect to the $T_2$ norm.
• Let $L_1$ be the sublist of $L_0$ of polynomials whose discriminant has minimal absolute value.
• For a polynomial $P = x^n + a_1x^{n-1} + \dots + a_n$, let $S(P) = (|a_1|,a_1,\dots,|a_n|,a_n)$, and order the polynomials in $L_1$ by the lexicographic order of the vectors $S(P)$.

Then the reduced defining polynomial of $K$ is the first polynomial in $L_1$ with respect to this order.

The pari/gp function polredabs() computes reduced defining polynomials, which are also commonly called polredabs polynomials.