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ON THE CYCLOTOMIC UNITS IN FUNCTION FIELDS

ON THE CYCLOTOMIC UNITS IN FUNCTION FIELDS
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On the distribution of the signs of cyclotomic units of a cyclotomic field

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Primary cyclotomic units and a proof of Catalans conjecture

Journal für die reine und angewandte Mathematik (Crelles Journal), 2004
Catalan's conjecture (1844) predicts that 8 and 9 are the only consecutive integers which are both perfect powers. This famous conjecture is proved in this paper. Catalan's conjecture corresponds to the Diophantine equation \(x^p-y^q=1\), where \(p\) and \(q\) are prime numbers and \(x\), \(y\) are rational integers with \(xy\ne 0\). The case of \(q=2\)
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Modular Units Inside Cyclotomic Units

The Annals of Mathematics, 1980
If H is an abelian extension of an imaginary quadratic field K, then the group of units of H contains an important subgroup consisting of modular units. These are obtained as special values of certain modular functions, and are analogous to the cyclotomic units of an abelian extension of Q.
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Cyclotomic units in z p -extensions

Israel Journal of Mathematics, 1991
LetK0 be the maximal real subfield of the field generated by thep-th root of 1 over ℚ, andK∞ be the basic Zp-extension ofK0 for a fixed odd primep. LetKn be itsn-th layer of this tower. For eachn, we denote the Sylowp-subgroup of the ideal class group ofKn byAn, and that ofEnCn byBn, whereEn (resp.Cn) is the group of units (resp. cyclotomic units ofKn.
Jae Moon Kim, Sunghan Bae, In-Sok Lee
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Cyclotomic unit and its fermat quotient

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 1997
In order to investigate the Leopoldt conjecture by an algebraic method, the author defined and studied the Fermat quotient and the level of a unit in a previous paper [cf. \textit{T. Shimada}, Tokyo Metropolitan University Mathematical Preprint Series, No. 5 (1996); see also his related paper in Acta Arith. 76, 335-358 (1996; Zbl 0867.11076)].
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Cyclotomic units over finite fields

Rendiconti del Circolo Matematico di Palermo, 1995
The paper focusses on determining the size \(S\) of the image of the group of all cyclotomic units in \(\mathbb{Q}(\zeta_p)^+\) after reducing \(\mathbb{Z}[\zeta_p]^+\) modulo a prime \(q\neq p\). Knowing \(S\), some indices which naturally come up when studying units in integral group rings \(\mathbb{Z}[C]\) of cyclic groups \(C\) of order \(pq\) (see
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DOUBLE COVERINGS AND UNIT SQUARE PROBLEMS FOR CYCLOTOMIC FIELDS

International Journal of Number Theory, 2011
In this paper, using the theory of double coverings of cyclotomic fields, we give a formula for [Formula: see text], where K = ℚ(ζn), G = Gal (K/ℚ), 𝔽2 = ℤ/2ℤ and UK is the unit group of K. We explicitly determine all the cyclotomic fields K = ℚ(ζn) such that [Formula: see text]. Then we apply it to the unit square problem raised in [Y.
Li, Yan, Ma, Lianrong
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Cyclotomic units and the unit group of an elementary abelian group ring

Archiv der Mathematik, 1985
Let A be a finite abelian group, and let U(A) be the group of units of \({\mathbb{Z}}A\) modulo torsion. Consider the maps \[ \prod_{C}U(C)\to^{\alpha}U(A)\to^{\beta}\prod_{K}U(K) \] where C and K run over the sets of cyclic subgroups and factor-groups of A, respectively.
Hoechsmann, K., Sehgal, S. K., Weiss, A.
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Cyclotomic Units

2014
Yuri F. Bilu, Yann Bugeaud, Maurice Mignotte   +2 more
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