Prime decomposition in the anti-cyclotomic extension
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- Math. Comp. 76 (2007), 2127-2138 Request permission
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.References
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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
- Email: brink@math.ku.dk
- Received by editor(s): October 21, 2005
- Received by editor(s) in revised form: July 4, 2006
- Published electronically: April 17, 2007
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 76 (2007), 2127-2138
- MSC (2000): Primary 11R32
- DOI: https://doi.org/10.1090/S0025-5718-07-01964-3
- MathSciNet review: 2336287