scholarly journals Power integral bases in prime-power cyclotomic fields

2006 ◽  
Vol 120 (2) ◽  
pp. 372-384 ◽  
Author(s):  
István Gaál ◽  
Leanne Robertson
Keyword(s):  
Prime Power ◽  
Cyclotomic Fields ◽  
Integral Bases

MONOGENEITY IN CYCLOTOMIC FIELDS

2010 ◽  
Vol 06 (07) ◽  
pp. 1589-1607 ◽  
Author(s):  
LEANNE ROBERTSON
Keyword(s):  
Unit Circle ◽  
Number Field ◽  
Complex Plane ◽  
Cyclotomic Field ◽  
Difficult Problem ◽  
Cyclotomic Fields ◽  
Ring Extension ◽  
Integral Basis ◽  
Ring Of Integers ◽  
Integral Bases

A number field is said to be monogenic if its ring of integers is a simple ring extension ℤ[α] of ℤ. It is a classical and usually difficult problem to determine whether a given number field is monogenic and, if it is, to find all numbers α that generate a power integral basis {1, α, α2, …, αk} for the ring. The nth cyclotomic field ℚ(ζn) is known to be monogenic for all n, and recently Ranieri proved that if n is coprime to 6, then up to integer translation all the integral generators for ℚ(ζn) lie on the unit circle or the line Re (z) = 1/2 in the complex plane. We prove that this geometric restriction extends to the cases n = 3k and n = 4k, where k is coprime to 6. We use this result to find all power integral bases for ℚ(ζn) for n = 15, 20, 21, 28. This leads us to a conjectural solution to the problem of finding all integral generators for cyclotomic fields.

scholarly journals Power integral bases for real cyclotomic fields

2005 ◽  
Vol 71 (1) ◽  
pp. 167-173 ◽  
Author(s):  
Laurel Miller-Sims ◽  
Leanne Robertson
Keyword(s):  
Cyclotomic Field ◽  
Cyclotomic Fields ◽  
Integral Basis ◽  
Algebraic Integers ◽  
Ring Of Integers ◽  
Root Of Unity ◽  
Integral Bases

We consider the problem of determining all power integral bases for the maximal real subfield Q (ζ + ζ−1) of the p-th cyclotomic field Q (ζ), where p ≥ 5 is prime and ζ is a primitive p-th root of unity. The ring of integers is Z[ζ+ζ−1] so a power integral basis always exists, and there are further non-obvious generators for the ring. Specifically, we prove that if or one of the Galois conjugates of these five algebraic integers. Up to integer translation and multiplication by −1, there are no additional generators for p ≤ 11, and it is plausible that there are no additional generators for p > 13 as well. For p = 13 there is an additional generator, but we show that it does not generalise to an additional generator for 13 < p < 1000.

scholarly journals Kummer's lemma for prime power cyclotomic fields

1992 ◽  
Vol 40 (2) ◽  
pp. 165-173 ◽  
Author(s):  
Lawrence C Washington
Keyword(s):  
Prime Power ◽  
Cyclotomic Fields

scholarly journals A NOTE ON THE EQUIVALENCE OF THE PARITY OF CLASS NUMBERS AND THE SIGNATURE RANKS OF UNITS IN CYCLOTOMIC FIELDS

2018 ◽  
Vol 238 ◽  
pp. 206-214 ◽  
Author(s):  
DAVID S. DUMMIT
Keyword(s):  
Prime Power ◽  
Cyclotomic Field ◽  
Class Numbers ◽  
Cyclotomic Fields ◽  
Circular Units

We collect some statements regarding equivalence of the parities of various class numbers and signature ranks of units in prime power cyclotomic fields. We correct some misstatements in the literature regarding these parities by providing an example of a prime cyclotomic field where the signature rank of the units and the signature rank of the circular units are not equal.

scholarly journals An infinite family of Williamson matrices

1977 ◽  
Vol 24 (2) ◽  
pp. 252-256 ◽  
Author(s):  
Edward Spence
Keyword(s):  
Prime Power ◽  
Hadamard Matrix ◽  
Infinite Family ◽  
Williamson Matrices

AbstractIn this paper the following result is proved. Suppose there exists a C-matrix of order n + 1. Then if n≡1 (mod 4) there exists a Hadamard matrix of order 2nr(n + 1), while if n≡3 (mod 4) there exists a Hadamard matrix of order nr(n + 1) for all r ≧0. If n≡1 (mod 4) is a prime power, the method is adapted to prove the existence of a Hadamard matrix of the Williamson type, of order 2nr(n + 1), for all r ≧0.

scholarly journals Certain locally nilpotent varieties of groups

2003 ◽  
Vol 67 (1) ◽  
pp. 115-119
Author(s):  
Alireza Abdollahi
Keyword(s):  
Prime Power ◽  
Power Exponent ◽  
Variety Of Groups ◽  
Periodic Groups ◽  
Locally Nilpotent

Let c ≥ 0, d ≥ 2 be integers and be the variety of groups in which every d-generator subgroup is nilpotent of class at most c. N.D. Gupta asked for what values of c and d is it true that is locally nilpotent? We prove that if c ≤ 2d + 2d−1 − 3 then the variety is locally nilpotent and we reduce the question of Gupta about the periodic groups in to the prime power exponent groups in this variety.

scholarly journals The energy of integral circulant graphs with prime power order

2011 ◽  
Vol 5 (1) ◽  
pp. 22-36 ◽  
Author(s):  
J.W. Sander ◽  
T. Sander
Keyword(s):  
Adjacency Matrix ◽  
Prime Power ◽  
Prime Power Order ◽  
Circulant Graph ◽  
Closed Formula ◽  
Circulant Graphs ◽  
Vertex Set

The energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd graphs. Such a graph can be characterized by its vertex count n and a set D of divisors of n such that its vertex set is Zn and its edge set is {{a,b} : a, b ? Zn; gcd(a-b, n)? D}. For an integral circulant graph on ps vertices, where p is a prime, we derive a closed formula for its energy in terms of n and D. Moreover, we study minimal and maximal energies for fixed ps and varying divisor sets D.

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