scholarly journals On the spectra of Pisot-cyclotomic numbers

2018 ◽  
Vol 108 (7) ◽  
pp. 1729-1756
Author(s):  
Kevin G. Hare ◽  
Zuzana Masáková ◽  
Tomáš Vávra
Keyword(s):  
Cyclotomic Numbers

On ℤ-Modules of Algebraic Integers

2009 ◽  
Vol 61 (2) ◽  
pp. 264-281 ◽  
Author(s):  
J. P. Bell ◽  
K. G. Hare
Keyword(s):  
Algebraic Integer ◽  
Full Rank ◽  
Pisot Number ◽  
List Type ◽  
Algebraic Integers ◽  
Pisot Numbers ◽  
Cyclotomic Numbers ◽  
Special Cases ◽  
Fixed Order ◽  
Fixed Positive Integer

Abstract. Let q be an algebraic integer of degree d ≥ 2. Consider the rank of the multiplicative subgroup of ℂ* generated by the conjugates of q. We say q is of full rank if either the rank is d − 1 and q has norm ±1, or the rank is d. In this paper we study some properties of ℤ[q] where q is an algebraic integer of full rank. The special cases of when q is a Pisot number and when q is a Pisot-cyclotomic number are also studied. There are four main results.(1)If q is an algebraic integer of full rank and n is a fixed positive integer, then there are only finitely many m such that disc `ℤ[qm]´ = disc `ℤ[qn]´.(2)If q and r are algebraic integers of degree d of full rank and ℤ[qn] = ℤ[rn] for infinitely many n, then either q = ωr′ or q = Norm(r)2/dω/r′ , where r ′ is some conjugate of r and ω is some root of unity.(3)Let r be an algebraic integer of degree at most 3. Then there are at most 40 Pisot numbers q such that ℤ[q] = ℤ[r].(4)There are only finitely many Pisot-cyclotomic numbers of any fixed order.

On Cyclotomic Numbers of Order Sixteen

1954 ◽  
Vol 6 ◽  
pp. 449-454 ◽  
Author(s):  
Emma Lehmer
Keyword(s):  
Linear Combination ◽  
Primitive Root ◽  
Number Of Solutions ◽  
Integer Coefficients ◽  
Cyclotomic Numbers ◽  
Image Position ◽  
Mod P

It has been shown by Dickson (1) that if (i, j)8 is the number of solutions of (mod p),then 64(i,j)8 is expressible for each i,j, as a linear combination with integer coefficients of p, x, y, a, and b where,anda ≡ b ≡ 1 (mod 4),while the sign of y and b depends on the choice of the primitive root g. There are actually four sets of such formulas depending on whether p is of the form 16n + 1 or 16n + 9 and whether 2 is a quartic residue or not.

scholarly journals A classification of (some) Pisot-Cyclotomic numbers

2005 ◽  
Vol 115 (2) ◽  
pp. 215-229 ◽  
Author(s):  
J.P. Bell ◽  
K.G. Hare
Keyword(s):  
Cyclotomic Numbers

scholarly journals Whitemanʼs generalized cyclotomic numbers with respect to t primes

2012 ◽  
Vol 18 (3) ◽  
pp. 634-644 ◽  
Author(s):  
Jing Cao ◽  
Qin Yue ◽  
Liqin Hu
Keyword(s):  
Cyclotomic Numbers

scholarly journals The cyclotomic numbers of order fourteen

1966 ◽  
Vol 11 (3) ◽  
pp. 263-279 ◽  
Author(s):  
Joseph Muskat
Keyword(s):  
Cyclotomic Numbers

scholarly journals The cyclotomic numbers of order eleven

1975 ◽  
Vol 26 (4) ◽  
pp. 365-383 ◽  
Author(s):  
Philip Leonard ◽  
Kenneth Williams
Keyword(s):  
Cyclotomic Numbers
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