On computing the discriminant of an algebraic number field
HTML articles powered by AMS MathViewer
- by Theresa P. Vaughan PDF
- Math. Comp. 45 (1985), 569-584 Request permission
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 $g({\mathbf {r}})/p \in \mathcal {R}$.References
- Ken Byrd and Theresa P. Vaughan, A group of integral points in a matrix parallelepiped, Linear Algebra Appl. 30 (1980), 155–166. MR 568788, DOI 10.1016/0024-3795(80)90191-3
- Harvey Cohn, A classical invitation to algebraic numbers and class fields, Universitext, Springer-Verlag, New York-Heidelberg, 1978. With two appendices by Olga Taussky: “Artin’s 1932 Göttingen lectures on class field theory” and “Connections between algebraic number theory and integral matrices”. MR 506156
- Kenneth Hoffman and Ray Kunze, Linear algebra, Prentice-Hall Mathematics Series, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1961. MR 0125849
- Morris Newman, Integral matrices, Pure and Applied Mathematics, Vol. 45, Academic Press, New York-London, 1972. MR 0340283
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Math. Comp. 45 (1985), 569-584
- MSC: Primary 11R29; Secondary 11Y40
- DOI: https://doi.org/10.1090/S0025-5718-1985-0804946-1
- MathSciNet review: 804946