Finding fundamental units in totally real fields

1984 ◽  
Vol 96 (2) ◽  
pp. 191-194 ◽  
Author(s):  
T. W. Cusick
Keyword(s):  
Cubic Field ◽  
Image Position ◽  
Totally Real ◽  
Totally Real Fields ◽  
Fundamental Units

Let F a totally real cubic field. For any number α in F, let α, α′, α″ denote the conjugates of α. Define the function T(α) by.

The determination of units in totally real cubic fields

1960 ◽  
Vol 56 (4) ◽  
pp. 318-321 ◽  
Author(s):  
H. J. Godwin
Keyword(s):  
Trial And Error ◽  
Integral Lattice ◽  
Cubic Field ◽  
Cubic Fields ◽  
Trial And Error Method ◽  
Totally Real ◽  
Rational Integral ◽  
Fundamental Units ◽  
Error Method

The determination of a pair of fundamental units in a totally real cubic field involves two operations—finding a pair of independent units (i.e. such that neither is a power of the other) and from these a pair of fundamental units (i.e. a pair ε1; ε2 such that every unit of the field is of the form with rational integral m, n). The first operation may be accomplished by exploring regions of the integral lattice in which two conjugates are small or else by factorizing small primes and comparing different factorizations—a trial-and-error method, but often a quick one. The second operation is accomplished by obtaining inequalities which must be satisfied by a fundamental unit and its conjugates and finding whether or not a unit exists satisfying these inequalities. Recently Billevitch ((1), (2)) has given a method, of the nature of an extension of the first method mentioned above, which involves less work on the second operation but no less on the first.

A note on Cusick's theorem on units in totally real cubic fields

1984 ◽  
Vol 95 (1) ◽  
pp. 1-2 ◽  
Author(s):  
H. J. Godwin
Keyword(s):  
Present Note ◽  
Cubic Field ◽  
Cubic Fields ◽  
Totally Real ◽  
Fundamental Units

Let ε = ε1, with conjugates ε2, ε3, be a unit in a totally real cubic field, and let . Let ε be a unit for which T (ε) is least and let η be a unit, not a power of ε, for which T(η) is least. It was shown by Cusick[l] that ε,η form a pair of fundamental units under certain conditions. The purpose of the present note is to show that these conditions are unnecessary and that ε, η form a pair of fundamental units in all cases.

Finding fundamental units in cubic fields

1982 ◽  
Vol 92 (3) ◽  
pp. 385-389 ◽  
Author(s):  
T. W. Cusick
Keyword(s):  
Computational Problem ◽  
Cubic Field ◽  
Cubic Fields ◽  
Totally Real ◽  
Fundamental Units

This paper improves a method of Godwin (4) for finding a pair of fundamental units in a totally real cubic field. The determination of such a unit pair is a well known computational problem. There is an old algorithm (circa 1896) of Voronoi which solves this problem, but the algorithm is quite complicated (an account of it is given in the book of Delone and Faddeev ((3), chapter IV, part A)). The method of Godwin is, in principle, much simpler. However, this method also has its drawbacks (more is said about this in Section 4 below). Indeed, when Godwin's student Angell produced his large table (see (1)) of totally real cubic fields some 15 years after (4) appeared, Voronoi's algorithm was used to compute the pairs of fundamental units.

scholarly journals Λ-adic modular symbols over totally real fields

2011 ◽  
pp. 841-865 ◽  
Author(s):  
Baskar Balasubramanyam ◽  
Matteo Longo
Keyword(s):  
Modular Symbols ◽  
Totally Real ◽  
Totally Real Fields

scholarly journals Motives over totally real fields and $p$-adic $L$-functions

1994 ◽  
Vol 44 (4) ◽  
pp. 989-1023 ◽  
Author(s):  
Alexei A. Panchishkin
Keyword(s):  
Totally Real ◽  
Totally Real Fields

scholarly journals Asymptotic generalized Fermat’s last theorem over number fields

2019 ◽  
Vol 16 (05) ◽  
pp. 907-924
Author(s):  
Yasemin Kara ◽  
Ekin Ozman
Keyword(s):  
Number Fields ◽  
Real Field ◽  
Quadratic Number ◽  
Fermat's Last Theorem ◽  
Fermat’S Last Theorem ◽  
Imaginary Quadratic Number ◽  
Fermat Equation ◽  
Totally Real ◽  
Totally Real Fields ◽  
Generalized Fermat Equation

Recent work of Freitas and Siksek showed that an asymptotic version of Fermat’s Last Theorem (FLT) holds for many totally real fields. This result was extended by Deconinck to the generalized Fermat equation of the form [Formula: see text], where [Formula: see text] are odd integers belonging to a totally real field. Later Şengün and Siksek showed that the asymptotic FLT holds over number fields assuming two standard modularity conjectures. In this work, combining their techniques, we show that the generalized Fermat’s Last Theorem (GFLT) holds over number fields asymptotically assuming the standard conjectures. We also give three results which show the existence of families of number fields on which asymptotic versions of FLT or GFLT hold. In particular, we prove that the asymptotic GFLT holds for a set of imaginary quadratic number fields of density 5/6.

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