On computing the discriminant of an algebraic number field

Abstract:

Let $f(x)$ be a monic irreducible polynomial in ${\mathbf {Z}}[x]$, and r a root of $f(x)$ in C. Let K be the field ${\mathbf {Q(r)}}$ and $\mathcal {R}$ the ring of integers in K. Then for some $k \in {\mathbf {Z}}$, $\operatorname {disc} {\mathbf {r}} = {k^2} \operatorname {disc} \mathcal {R}$ . In this paper we give constructive methods for (a) deciding if a prime p divides k, and (b) if $p|k$, finding a polynomial $g(x) \in {\mathbf {Z}}[x]$ so that $g(x)\nequiv 0\;\pmod p$ but $g({\mathbf {r}})/p \in \mathcal {R}$.