scholarly journals The number of different lengths of irreducible factorization of a natural number in an algebraic number field

1980 ◽  
Vol 36 (1) ◽  
pp. 59-86 ◽  
Author(s):  
S. Allen ◽  
P. Pleasants
Keyword(s):  
Natural Number ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Irreducible Factorization

scholarly journals On the maximal abelian ℓ-extension of a finite algebraic number field with given ramification

1978 ◽  
Vol 70 ◽  
pp. 183-202 ◽  
Author(s):  
Hiroo Miki
Keyword(s):  
Natural Number ◽  
Galois Group ◽  
Prime Number ◽  
Number Field ◽  
Algebraic Number ◽  
Class Number ◽  
Algebraic Number Field

Let k be a finite algebraic number field and let ℓ be a fixed odd prime number. In this paper, we shall prove the equivalence of certain rather strong conditions on the following four things (1) ~ (4), respectively : (1) the class number of the cyclotomic Zℓ-extension of k,(2) the Galois group of the maximal abelian ℓ-extension of k with given ramification,(3) the number of independent cyclic extensions of k of degree ℓ, which can be extended to finite cyclic extensions of k of any ℓ-power degree, and(4) a certain subgroup Bk(m, S) (cf. § 2) of k×/k×)ℓm for any natural number m (see the main theorem in §3).

Algebraic extensions in nonstandard models and Hilbert's irreducibility theorem

1988 ◽  
Vol 53 (2) ◽  
pp. 470-480 ◽  
Author(s):  
Masahiro Yasumoto
Keyword(s):  
Natural Number ◽  
Prime Number ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Algebraic Extension ◽  
Algebraic Integers ◽  
Nonstandard Models ◽  
Ring Of Algebraic Integers

LetKbe an algebraic number field andIKthe ring of algebraic integers inK. *Kand *IKdenote enlargements ofKandIKrespectively. LetxЄ *K–K. In this paper, we are concerned with algebraic extensions ofK(x)within *K. For eachxЄ *K–Kand each natural numberd, YK(x,d)is defined to be the number of algebraic extensions ofK(x)of degreedwithin *K.xЄ *K–Kis called a Hilbertian element ifYK(x,d)= 0 for alldЄ N,d> 1; in other words,K(x)has no algebraic extension within *K. In their paper [2], P. C. Gilmore and A. Robinson proved that the existence of a Hilbertian element is equivalent to Hilbert's irreducibility theorem. In a previous paper [9], we gave many Hilbertian elements of nonstandard integers explicitly, for example, for any nonstandard natural numberω, 2ωPωand 2ω(ω3+ 1) are Hilbertian elements in*Q, where pωis theωth prime number.

scholarly journals On the conjugates of certain algebraic integers

2016 ◽  
Vol 99 (113) ◽  
pp. 281-285 ◽  
Author(s):  
Toufik Zaïmi
Keyword(s):  
Natural Number ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Pisot Number ◽  
Algebraic Integers ◽  
Pisot Numbers

A well-known theorem, due to C. J. Smyth, asserts that two conjugates of a Pisot number, having the same modulus are necessary complex conjugates. We show that this result remains true for K-Pisot numbers, where K is a real algebraic number field. Also, we prove that a j-Pisot number, where j is a natural number, can not have more than 2j conjugates with the same modulus.

On correspondence between orders and ideals with a normal basis in cyclic subfields of Q(ξp) of a prime degree l

2010 ◽  
Vol 60 (6) ◽  
Author(s):  
Juraj Kostra
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Normal Basis ◽  
Prime Degree ◽  
One To One

AbstractLet K be a tamely ramified cyclic algebraic number field of prime degree l. In the paper one-to-one correspondence between all orders of K with a normal basis and all ideals of K with a normal basis is given.

scholarly journals Automorphy factors for a Hilbert modular group

1988 ◽  
Vol 30 (2) ◽  
pp. 231-236
Author(s):  
Shigeaki Tsuyumine
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Rational Number ◽  
Modular Group ◽  
Algebraic Number Field ◽  
Maximal Order ◽  
Hilbert Modular Group ◽  
Hilbert Modular ◽  
Totally Real

Let K be a totally real algebraic number field of degree n > 1, and let OK be the maximal order. We denote by гk, the Hilbert modular group SL2(OK) associated with K. On the extent of the weight of an automorphy factor for гK, some restrictions are imposed, not as in the elliptic modular case. Maass [5] showed that the weight is integral for K = ℚ(√5). It was shown by Christian [1] that for any Hilbert modular group it is a rational number with the bounded denominator depending on the group.

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