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Mathematics of Computation

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Prime decomposition in the anti-cyclotomic extension

Author: David Brink
Journal: Math. Comp. 76 (2007), 2127-2138
MSC (2000): Primary 11R32
Published electronically: April 17, 2007
MathSciNet review: 2336287
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Abstract: For an imaginary quadratic number field $ K$ and an odd prime number $ l$, the anti-cyclotomic $ \mathbb{Z}_l$-extension of $ K$ is defined. For primes $ \mathfrak{p}$ of $ K$, decomposition laws for $ \mathfrak{p}$ in the anti-cyclotomic extension are given. We show how these laws can be applied to determine if the Hilbert class field (or part of it) of $ K$ is $ \mathbb{Z}_l$-embeddable. For some $ K$ and $ l$, we find explicit polynomials whose roots generate the first step of the anti-cyclotomic extension and show how the prime decomposition laws give nice results on the splitting of these polyniomials modulo $ p$. The article contains many numerical examples.

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Additional Information

David Brink
Affiliation: Department of Mathematics, Universitetsparken 5, 2100 Copenhagen, Denmark
Address at time of publication: Departamento de Matemática, Universidade de Brasília, 70910-900 Brasília-DF-Brazil

Keywords: Prime decomposition, imaginary quadratic number fields, ring class fields, pro-cyclic $l$-extensions, factorisation of polynomials modulo $p$.
Received by editor(s): October 21, 2005
Received by editor(s) in revised form: July 4, 2006
Published electronically: April 17, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.