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 $P = x^n + a_1x + \dots + a_n$, 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.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by David Roe on 2020-10-13 15:55:08

**Referred to by:**

**History:**(expand/hide all)

- 2020-10-13 15:55:08 by David Roe (Reviewed)
- 2020-05-01 08:25:55 by John Cremona
- 2018-07-07 21:45:55 by Alina Bucur (Reviewed)

**Differences**(show/hide)