Integral Bases for Subfields of Cyclotomic Fields

1997 ◽  
Vol 8 (4) ◽  
pp. 279-289
Author(s):  
Thomas Breuer
Keyword(s):  
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 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

A Note to the Paper on Integral Bases

1958 ◽  
Vol 9 (1) ◽  
pp. 173
Author(s):  
Virginia Hanly ◽  
H. B. Mann
Keyword(s):  
Integral Bases

scholarly journals A GENERALISED KUMMER'S CONJECTURE

2010 ◽  
Vol 52 (3) ◽  
pp. 453-472 ◽  
Author(s):  
M. J. R. MYERS
Keyword(s):  
Natural Number ◽  
Class Numbers ◽  
Cyclotomic Fields ◽  
Rate Of Growth ◽  
Relative Class ◽  
Almost All

AbstractKummer's conjecture predicts the rate of growth of the relative class numbers of cyclotomic fields of prime conductor. We extend Kummer's conjecture to cyclotomic fields of conductor n, where n is any natural number. We show that the Elliott–Halberstam conjecture implies that this generalised Kummer's conjecture is true for almost all n but is false for infinitely many n.

Power Integral Bases in Composits of Number Fields

1998 ◽  
Vol 41 (2) ◽  
pp. 158-165 ◽  
Author(s):  
István Gaál
Keyword(s):  
Number Fields ◽  
Parametric Family ◽  
Prime Degree ◽  
Index Form ◽  
Form Equation ◽  
Integral Bases ◽  
Quadratic Extensions ◽  
Totally Real

AbstractIn the present paper we consider the problem of finding power integral bases in number fields which are composits of two subfields with coprime discriminants. Especially, we consider imaginary quadratic extensions of totally real cyclic number fields of prime degree. As an example we solve the index form equation completely in a two parametric family of fields of degree 10 of this type.

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