Spectres euclidiens et inhomogènes des corps de nombres

Jean-Paul Cerri 1, 2
1 SPACES - Solving problems through algebraic computation and efficient software
LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications, INRIA Lorraine
Abstract : This thesis has a twofold purpose. Firstly, it attempts to address various issues relating to the concepts of Euclidean and inhomogeneous spectra (for the norm form), especially those relating to the Euclidean minimum of a number field (for the norm form). We establish, in particular, that for every number field K, the Euclidean minimum of K, denoted by M(K), and the inhomogeneous minimum of K, M(\overline{K}), are equal, and that, if the unit rank of K is strictly greater than 1, the Euclidean and inhomogeneous spectra of K are equal and rational when K is not CM. The results that we have established in the latter case, have as a consequence, the decidability of whether K is norm-euclidean of not.
We also show how to explicitly compute M(K). We present an algorithm for cases where K is a totally real number field. This algorithm has enabled us to establish tables up to degree 8, and it
may be transposed to any number field. Moreover, this algorithm has enabled us to find many examples of number fields with class number 1, which are not norm-Euclidean but m-stage norm-Euclidean for m=2.
Document type :
Theses
Mathematics [math]. Université Henri Poincaré - Nancy I, 2005. French


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Submitted on : Monday, December 5, 2005 - 7:42:56 PM
Last modification on : Tuesday, December 6, 2005 - 11:56:06 AM

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Jean-Paul Cerri. Spectres euclidiens et inhomogènes des corps de nombres. Mathematics [math]. Université Henri Poincaré - Nancy I, 2005. French. <tel-00011151>

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