Mathematics of Computation

Author(s): Jean-Paul Cerri.
Journal: Math. Comp. 76 (2007), 1547-1575.
MSC (2000): Primary 11Y40; Secondary 11R04, 12J15, 13F07
Posted: February 27, 2007
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Abstract: This article deals with the determination of the Euclidean minimum

of a totally real number field

of degree $n\geq 2$ , using techniques from the geometry of numbers. Our improvements of existing algorithms allow us to compute Euclidean minima for fields of degree

and small discriminants, most of which were previously unknown. Tables are given at the end of this paper.

[A2X]: THE A2X LABORATORY, Number field tables available from ftp://megrez.math.u-bordeaux.fr/pub/numberfields.
[BSD]: E.S. BARNES AND H.P.F. SWINNERTON-DYER, The inhomogeneous minima of binary quadratic forms, I, Acta Mathematica 87 (1952), 259-323. II, ibid. 88 (1952), 279-316. MR 0053162 (14:730a); MR 0054654 (14:956a)
[Ca]: J.W.S. CASSELS, An introduction to the geometry of numbers, Classics in Mathematics, Springer-Verlag, 1971. MR 0306130 (46:5257)
[Ce1]: J-P. CERRI, De l'euclidianité de $\mathbb{Q} \left(\sqrt{2+\sqrt{2+\sqrt{2}}}\right)$ et $\mathbb{Q}\left(\sqrt{2+\sqrt{2}}\right)$ pour la norme, J. Th. Nombres Bordeaux 12 (2000), 103-126. MR 1827842 (2002f:11145)
[Ce2]: J-P. CERRI, Euclidean and inhomogeneous spectra of number fields with unit rank greater than (to appear in Journal für die Reine und Angewandte Mathematik).
[Ce3]: J-P. CERRI, Spectres euclidiens et inhomogènes des corps de nombres, Thèse de Doctorat, Université Henri Poincaré, Nancy (2005) available from http://tel.ccsd.cnrs.fr/tel-00011151.
[Ce4]: J-P. CERRI, Tables of Euclidean minima of totally real number fields, available from ftp://megrez.math.u-bordeaux.fr/pub/cerri
[CL]: S. CAVALLAR AND F. LEMMERMEYER, The Euclidean algorithm in cubic number fields, in Gyory, Petho, Sos, eds., Proceedings Number Theory Eger 1996, de Gruyter, 1998, 123-146. MR 1628838 (99e:11137)
[CD]: H. COHN AND J. DEUTSCH, Use of a computer scan to prove $\mathbb{Q}\left(\sqrt{2+\sqrt{2}}\right)$ and $\mathbb{Q} \left(\sqrt{3+\sqrt{2}}\right)$ are Euclidean, Mathematics of Computation 46 (1986), 295-299. MR 0815850 (87b:11105)
[K]: J. KLÜNERS, Tables available at http://www.mathematik.uni-kassel.de/~klueners
[L]: F. LEMMERMEYER, The Euclidean algorithm in algebraic number fields, Expositiones Mathematicae 13 (1995), 385-416. MR 1362867 (96i:11115)
[P]: C. BATUT, K. BELABAS, D. BERNARDI, H. COHEN, M. OLIVIER, The Pari/GP system, http://pari.math.u-bordeaux.fr
[Q]: R. QUÊME, A computer algorithm for finding new Euclidean number fields, J. Th. Nombres Bordeaux 10 (1998), 33-48. MR 1827284 (2002g:11182)
[T]: W.T. TUTTE, Graph Theory, Encyclopedia of Mathematics and its Applications, vol. 21, Addison-Wesley, Reading, MA, 1984. MR 0746795 (87c:05001)

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