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By the primitive element theorem, a number field $K$ can always be represented as $K = \Q[X]/P(x)$ for some monic $P\in\Z[X]$. The pari/gp function polredabs provides a canonical such polynomial, defined as follows. 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 characteristic polynomials of primitive elements of $K$ that are in the ring of integers $\Z_K$ and minimal for the $T_2$ norm.
  • Let $L_1$ be the sublist of $L_0$ of polynomials whose discriminant has minimal absolute value.
  • Let $L_2$ be the sublist of $L_1$ defined as follows. In each pair $\{P(x), (-1)^{\deg P}P(-x)\}$ of monic polynomials, choose the unique polynomial $Q$ such that when written $Q(x) = Q_1(x^2)+xQ_2(x^2)$, at most one of $Q_1,Q_2$ has positive leading coefficient.
  • Order the integers by absolute value first, breaking ties by putting $|x|$ before $-|x|$. Order the polynomials in $L_2$ by the lexicographic order of the coefficient vectors (from the leading coefficient to the constant coefficient) where integers are ordered as above. Then the polredabs polynomial of $K$ is the smallest polynomial in $L_2$ for that order.
Knowl status:
  • Review status: reviewed
  • Last edited by Alina Bucur on 2018-07-07 21:45:55
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