scholarly journals A PRECISE RESULT ON THE ARITHMETIC OF NON-PRINCIPAL ORDERS IN ALGEBRAIC NUMBER FIELDS

2012 ◽  
Vol 11 (05) ◽  
pp. 1250087 ◽  
Author(s):  
ANDREAS PHILIPP
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Theoretical Approach ◽  
Class Group ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Algebraic Number Fields ◽  
Precise Result ◽  
Principal Order

Let R be an order in an algebraic number field. If R is a principal order, then many explicit results on its arithmetic are available. Among others, R is half-factorial if and only if the class group of R has at most two elements. Much less is known for non-principal orders. Using a new semigroup theoretical approach, we study half-factoriality and further arithmetical properties for non-principal orders in algebraic number fields.

scholarly journals On unramified cyclic extensions of degree l of algebraic number fields of degree l

1987 ◽  
Vol 107 ◽  
pp. 135-146 ◽  
Author(s):  
Yoshitaka Odai
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Preceding Paper ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Maximal Extension ◽  
Algebraic Number Fields ◽  
Cubic Field ◽  
Genus Field ◽  
Abelian Number Field

Let I be an odd prime number and let K be an algebraic number field of degree I. Let M denote the genus field of K, i.e., the maximal extension of K which is a composite of an absolute abelian number field with K and is unramified at all the finite primes of K. In [4] Ishida has explicitly constructed M. Therefore it is of some interest to investigate unramified cyclic extensions of K of degree l, which are not contained in M. In the preceding paper [6] we have obtained some results about this problem in the case that K is a pure cubic field. The purpose of this paper is to extend those results.

The discriminant of compositum of algebraic number fields

2019 ◽  
Vol 15 (02) ◽  
pp. 353-360
Author(s):  
Sudesh K. Khanduja
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Sufficient Conditions ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Rational Numbers ◽  
Necessary And Sufficient Conditions ◽  
Algebraic Number Fields ◽  
Necessary And Sufficient

For an algebraic number field [Formula: see text], let [Formula: see text] denote the discriminant of an algebraic number field [Formula: see text]. It is well known that if [Formula: see text] are algebraic number fields with coprime discriminants, then [Formula: see text] are linearly disjoint over the field [Formula: see text] of rational numbers and [Formula: see text], [Formula: see text] being the degree of [Formula: see text] over [Formula: see text]. In this paper, we prove that the converse of this result holds in relative extensions of algebraic number fields. We also give some more necessary and sufficient conditions for the analogue of the above equality to hold for algebraic number fields [Formula: see text] linearly disjoint over [Formula: see text].

scholarly journals On central extensions of a Galois extension of algebraic number fields

1984 ◽  
Vol 93 ◽  
pp. 133-148 ◽  
Author(s):  
Katsuya Miyake
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Algebraic Number ◽  
Galois Extension ◽  
Central Extension ◽  
Algebraic Closure ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Algebraic Number Fields ◽  
Central Extensions

Let k be an algebraic number field of finite degree, and K a finite Galois extension of k. A central extension L of K/k is an algebraic number field which contains K and is normal over k, and whose Galois group over K is contained in the center of the Galois group Gal(L/k). We denote the maximal abelian extensions of k and K in the algebraic closure of k by kab and Kab respectively, and the maximal central extension of K/k by MCK/k. Then we have Kab⊃MCK/k⊃kab·K.

scholarly journals A note on the characterization of CM-fields

1978 ◽  
Vol 26 (1) ◽  
pp. 26-30 ◽  
Author(s):  
P. E. Blanksby ◽  
J. H. Loxton
Keyword(s):  
Unit Circle ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Subject Classification ◽  
Algebraic Number Fields

AbstractThis note deals with some properties of algebraic number fields generated by numbers having all their conjugates on a circle. In particular, it is shown that an algebraic number field is a CM-field if and only if it is generated over the rationals by an element (not equal to ±1) whose conjugate all lie on the unit circle.Subject classification (Amer. Math. Soc. (MOS) 1970): 12 A 15, 12 A 40, 14 K 22.

An Independent System of Units in Certain Algebraic Number Fields

1985 ◽  
Vol 37 (4) ◽  
pp. 644-663
Author(s):  
Claude Levesque
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Algebraic Number Fields ◽  
Independent System ◽  
System Of Units ◽  
Image Position

For Kn = Q(ω) a real algebraic number field of degree n over Q such thatwith D ∊ N, d ∊ Z, d|D2, and D2 + 4d > 0, we proved in [5] (by using the approach of Halter-Koch and Stender [6]) that ifwiththenis an independent system of units of Kn.

scholarly journals The Genus Field and Genus Number in Algebraic Number Fields

1967 ◽  
Vol 29 ◽  
pp. 281-285 ◽  
Author(s):  
Yoshiomi Furuta
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Abelian Extension ◽  
Normal Extension ◽  
Finite Degree ◽  
Algebraic Number Fields ◽  
Genus Field ◽  
Genus Number

Let k be an algebraic number field and K be its normal extension of finite degree. Then the genus field K* of K over k is defined as the maximal unramified extension of K which is obtained from K by composing an abelian extension over k2). We call the degree (K*: K) the genus number of K over k.

Waring's problem in algebraic number fields

1961 ◽  
Vol 57 (3) ◽  
pp. 449-459 ◽  
Author(s):  
B. J. Birch
Keyword(s):  
Positive Integer ◽  
Number Field ◽  
Algebraic Number ◽  
Good Deal ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Singular Series ◽  
Algebraic Number Fields ◽  
Extra Information ◽  
Totally Positive

Let K be a finite algebraic number field, of degree R. Then those integers of K which may be expressed as a sum of dth powers generate a subring JK, d of the integers of K (JK, d need not be an ideal of K, as the simplest example K = Q(i), d = 2 shows. JK, d is an order, it in fact contains all integer multiples of d!; it also contains all rational integers). Siegel(12) showed that every sufficiently large totally positive integer of JK, d is the sum of at most (2d−1 + R) Rd totally positive dth powers; and he conjectured that the number of dth powers necessary should be independent of the field K—for instance, he had proved(11) that five squares are enough for every K. In this paper, we will show that, as far as the analytic part of the argument is concerned, Siegel's conjecture is correct. I have not been able to deal properly with the problem of proving that the singular series is positive; but since Siegel wrote, a good deal of extra information about singular series has been obtained, in particular by Stemmler(14) and Gray(4). The most spectacular consequence of all this is that if p is prime, then every large enough totally positive integer of JK, p is a sum of (2p + 1) totally positive pth. powers.

scholarly journals Leopoldt kernels and central extensions of algebraic number fields

1990 ◽  
Vol 120 ◽  
pp. 67-76 ◽  
Author(s):  
Katsuya Miyake
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Unit Group ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Prime Divisors ◽  
Finite Degree ◽  
Algebraic Number Fields ◽  
Central Extensions ◽  
The Real

Let k be an algebraic number field of finite degree, and p be a fixed rational prime. We denote the set of all the non-Archimedian prime divisors of k by S0(k) and the set of all the real Archimedian ones by (k). Put and S = S0 ∪ S∞, and define a subgroup of the unit group (k) of k by

Cubic forms over algebraic number fields

1969 ◽  
Vol 66 (2) ◽  
pp. 323-333 ◽  
Author(s):  
C. Ryavec
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Algebraic Number Fields ◽  
Rational Field ◽  
Cubic Form ◽  
Proper Subvariety ◽  
Almost All ◽  
Cubic Forms

In 1935 Tartakowski (7) proved that, in general, a cubic form in sufficiently many variables with coefficients in an algebraic number field K has a non-trivial zero in that field; and in the case when K is the rational field 57 variables suffice. Here, ‘in general’ means that the coefficients of the form do not lie in a proper subvariety of the coefficient space. Hence, Tartakowski's result holds for almost all cubic forms. Later, Lewis (5) proved that if K is any algebraic number field such that [K: Q] = n, then there exists a function ψ(n) such that every cubic form over K in m ≥ ψ(n) variables has a non-trivial zero in K. His bound, ψ(n), is extremely large; e.g. when K is the rational field, ψ(1) > 500.

scholarly journals On the integer ring of the compositum of algebraic number fields

1980 ◽  
Vol 77 ◽  
pp. 25-31 ◽  
Author(s):  
Toshitaka Kataoka
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Finite Extension ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Finite Degree ◽  
Algebraic Number Fields ◽  
Integer Ring

Let k be an algebraic number field of finite degree. For a finite extension L/k we denote by L/k the different of L/k, and by L the integer ring of L. Let K1 and K2 be finite extensions of k.

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