Computation of a power integral basis of a pure cubic number field

2007 ◽  
Vol 2 ◽  
pp. 601-606 ◽  
Author(s):  
L. El Fadil
Keyword(s):  
Number Field ◽  
Integral Basis

On a Theorem of Kawamoto on Normal Bases of Rings of Integers, II

2005 ◽  
Vol 48 (4) ◽  
pp. 576-579 ◽  
Author(s):  
Humio Ichimura
Keyword(s):  
Prime Number ◽  
Number Field ◽  
Class Group ◽  
Natural Homomorphism ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Integral Basis ◽  
Rings Of Integers ◽  
Integer Ring ◽  
The Ideal

AbstractLet m = pe be a power of a prime number p. We say that a number field F satisfies the property when for any a ∈ F×, the cyclic extension F(ζm, a1/m)/F(ζm) has a normal p-integral basis. We prove that F satisfies if and only if the natural homomorphism is trivial. Here K = F(ζm), and denotes the ideal class group of F with respect to the p-integer ring of F.

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.

A Number Field Without Any Integral Basis

1979 ◽  
Vol 52 (4) ◽  
pp. 248-251 ◽  
Author(s):  
Hugh M. Edgar
Keyword(s):  
Number Field ◽  
Integral Basis

A Number Field without Any Integral Basis

1979 ◽  
Vol 52 (4) ◽  
pp. 248 ◽  
Author(s):  
Hugh M. Edgar
Keyword(s):  
Number Field ◽  
Integral Basis

A Number Field Without a Relative Integral Basis

1971 ◽  
Vol 78 (8) ◽  
pp. 882-883 ◽  
Author(s):  
Robert MacKenzie ◽  
John Scheuneman
Keyword(s):  
Number Field ◽  
Integral Basis

A Number Field Without a Relative Integral Basis

1971 ◽  
Vol 78 (8) ◽  
pp. 882 ◽  
Author(s):  
Robert MacKenzie ◽  
John Scheuneman
Keyword(s):  
Number Field ◽  
Integral Basis

K-STRATUM LIMIT AND K-STRATUM INTEGRAL ON THE GENERALIZED NUMBER FIELD

1982 ◽  
Vol 2 (4) ◽  
pp. 375-388
Author(s):  
Jiwu Wang ◽  
Tai Kang
Keyword(s):  
Number Field
Sign in / Sign up

Export Citation Format

Share Document