Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0739.11023
Lecacheux, Odile
Units of a family of fields related to the curve $X\sb 1(25)$. (Unités d'une famille de corps liés à la courbe $X\sb 1(25)$.) (French)
[J] Ann. Inst. Fourier 40, No.2, 237-253 (1990). ISSN 0373-0956; ISSN 1777-5310/e

In 1974 Shanks introduced what he called the ``simplest cubic fields''. These were the cyclic extensions of degree three generated by the roots of $x\sp 3-tx\sp 2-(t+3)x-1$, where $t$ is an integer. Later M.-N. Gras found an analogous family of fields of degree four and E. Lehmer provided an example of such a family of degree five. The main interest in these fields resides in our ability, under certain conditions, to determine a fundamental system of units and to bound (or even evaluate) the class number.\par In a previous paper the author used the modular curve $X\sb 1(13)$ to construct a family of cyclic fields of degree 6, and in the paper under review he applies these same ideas, using the modular curve $X\sb 1(25)$, to construct a family of cyclic fields of degree 10, a family that was studied by Lehmer and by Schoof and Washington. It is shown how, under certain conditions, a fundamental system of units can be found for these fields. Also, in a particular case the connection between these units and the Gaussian periods of these Abelian extensions is described.
[H.C.Williams (Winnipeg)]

MSC 2000:: *11G16 Elliptic and modular units
11R27 Units and factorization
11R20 Other abelian and metabelian extensions
11H06 Lattices and convex bodies (number theoretic results)

Keywords: modular units; unit group; simplest cubic fields; cyclic extensions of degree three; fundamental system of units; cyclic fields of degree 10; Gaussian periods; Abelian extensions

Citations: Zbl 0664.12004

Cited in: Zbl 1172.11039 Zbl 1157.11014 Zbl 0809.11068

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]