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Mathematics of Computation

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On computing the discriminant of an algebraic number field

Author: Theresa P. Vaughan
Journal: Math. Comp. 45 (1985), 569-584
MSC: Primary 11R29; Secondary 11Y40
MathSciNet review: 804946
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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\vert 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 [Enhancements On Off] (What's this?)

  • [1] Ken Byrd & Theresa P. Vaughan, "A group of integral points in a matrix parallelepiped," Linear Algebra Appl., v. 30, 1980, pp. 155-166. MR 568788 (81f:15027)
  • [2] Harvey Cohn, A Classical Invitation to Algebraic Numbers and Class Fields, Springer-Verlag, Berlin and New York, 1978. MR 506156 (80c:12001)
  • [3] Kenneth Hoffman & Ray Kunze, Linear Algebra, Prentice-Hall, Englewood Cliffs, N. J., 1961. MR 0125849 (23:A3146)
  • [4] Morris Newman, Integral Matrices, Academic Press, New York, 1972. MR 0340283 (49:5038)

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Article copyright: © Copyright 1985 American Mathematical Society

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