A combinatorial problem in Abelian groups

1963 ◽  
Vol 59 (3) ◽  
pp. 559-562 ◽  
Author(s):  
Kenneth Rogers
Keyword(s):  
Prime Ideal ◽  
Abelian Groups ◽  
Combinatorial Problem ◽  
Algebraic Number Field ◽  
Prime Ideals ◽  
Ideal Class ◽  
Finite Abelian Groups ◽  
Ring Of Integers ◽  
Identity Class ◽  
The Ideal

Let α be a prime element of the ring of integers of an algebraic number field, R. Mr C. Sudler verbally raised the question as to how many prime ideal factors α can have. This is equivalent to a problem on the group of ideal classes of R, as we now show. If there is a prime α = p1p2 … Pk, where the prime ideal pi, is in the ideal class Pi, then P1P2…Pk equals the identity class, but no subproduct has this property. The converse holds since every ideal class contains prime ideals. So the problem is equivalent to one on finite Abelian groups, which we now write additively.

scholarly journals On nilpotent factors of congruent ideal class groups of Galois extensions

1976 ◽  
Vol 62 ◽  
pp. 13-28 ◽  
Author(s):  
Yoshiomi Furuta
Keyword(s):  
Class Group ◽  
Algebraic Number Field ◽  
Augmentation Ideal ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Class Groups ◽  
Finite Degree ◽  
Ring Of Integers ◽  
Ideal Class Groups ◽  
The Ideal

Let K be a Galois extension of an algebraic number field k of finite degree with Galois group g. Then g acts on a congruent ideal class group of K as a group of automorphisms, when the class field M over K corresponding to is normal over K. Let Ig be the augmentation ideal of the group ring Zg over the ring of integers Z, namely Ig be the ideal of Zg generated by σ − 1, σ running over all elements of g. Then is the group of all elements aσ-1 where a and σ belong to and g respectively.

scholarly journals Fast heuristic algorithms for computing relations in the class group of a quadratic order, with applications to isogeny evaluation

2016 ◽  
Vol 19 (A) ◽  
pp. 371-390 ◽  
Author(s):  
Jean-François Biasse ◽  
Claus Fieker ◽  
Michael J. Jacobson
Keyword(s):  
Class Group ◽  
Heuristic Algorithms ◽  
Large Degree ◽  
Prime Ideals ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Improved Performance ◽  
Novel Algorithms ◽  
Almost All ◽  
The Ideal

In this paper, we present novel algorithms for finding small relations and ideal factorizations in the ideal class group of an order in an imaginary quadratic field, where both the norms of the prime ideals and the size of the coefficients involved are bounded. We show how our methods can be used to improve the computation of large-degree isogenies and endomorphism rings of elliptic curves defined over finite fields. For these problems, we obtain improved heuristic complexity results in almost all cases and significantly improved performance in practice. The speed-up is especially high in situations where the ideal class group can be computed in advance.

scholarly journals Distribution of units of real quadratic number fields

2000 ◽  
Vol 158 ◽  
pp. 167-184 ◽  
Author(s):  
Yen-Mei J. Chen ◽  
Yoshiyuki Kitaoka ◽  
Jing Yu
Keyword(s):  
Prime Ideal ◽  
Quadratic Field ◽  
Number Fields ◽  
Prime Ideals ◽  
Positive Density ◽  
Real Quadratic Field ◽  
Quadratic Number ◽  
Ring Of Integers ◽  
Group Of Units ◽  
Real Quadratic

AbstractLet k be a real quadratic field and k, E the ring of integers and the group of units in k. Denoting by E() the subgroup represented by E of (k/)× for a prime ideal , we show that prime ideals for which the order of E() is theoretically maximal have a positive density under the Generalized Riemann Hypothesis.

On a problem in algebraic number theory

1996 ◽  
Vol 119 (2) ◽  
pp. 191-200 ◽  
Author(s):  
J. Wójcik
Keyword(s):  
Number Theory ◽  
Prime Ideal ◽  
Number Field ◽  
Algebraic Number ◽  
Multiplicative Group ◽  
Algebraic Number Field ◽  
Algebraic Number Theory ◽  
Ring Of Integers

Let K be an algebraic number field. If q is a prime ideal of the ring of integers of K and α is a number of K prime to q then Mq(α) denotes the multiplicative group generated by α modulo q. In the paper [5] there is the remark: ‘We do not know whether for all a, b, c ∈ ℚ with abc ≠ 0, |a| ≠ 1,|b| ≠ 1,|c| ≠ 1 there exist infinitely many primes q with Mq (a) = Mq (b) = Mq (c).’

scholarly journals Note on class number factors and prime decompositions

1977 ◽  
Vol 66 ◽  
pp. 167-182 ◽  
Author(s):  
Yoshiomi Furuta
Keyword(s):  
Algebraic Number ◽  
Class Number ◽  
Galois Extension ◽  
Class Group ◽  
Algebraic Number Field ◽  
Augmentation Ideal ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Finite Degree ◽  
Ring Of Integers

Let K be a Galois extension of an algebraic number field k of finite degree with Galois group g, be a congruent ideal class group of K, and M be the class field over K corresponding to . Assume that M is normal over k. Then g acts on as a group of automorphisms. Donote by lg the augmentation ideal of the group ring Zg over the ring of integers Z.

Bases for the Prime Ideals Associated with Certain Classes of Algebraic Varieties

1943 ◽  
Vol 39 (1) ◽  
pp. 35-48 ◽  
Author(s):  
L. S. Goddard
Keyword(s):  
Prime Ideal ◽  
Finite Basis ◽  
Ideal Theory ◽  
Prime Ideals ◽  
Algebraic Varieties ◽  
Twisted Cubic ◽  
Cubic Curve ◽  
Principal Theorem ◽  
General Method ◽  
The Ideal

The fact that the prime ideal associated with a given irreducible algebraic variety has a finite basis is a pure existence theorem. Only in a few isolated particular cases has the base for the ideal been found, and there appears to be no general method for determining the base which can be carried out in practice. Hilbert, who initiated the theory, proved that the prime ideal defining the ordinary twisted cubic curve has a base consisting of three quadrics, and contributions to the ideal theory of algebraic varieties have been made by König, Lasker, Macaulay and, more recently, by Zariski. A good summary, from the viewpoint of a geometer, is given by Bertini [(1), Chapter XII]. However, the tendency has been towards the development of the pure theory. In the following paper we actually find the bases for the prime ideals associated with certain classes of algebraic varieties. The paper falls into two parts. In Part I there is proved a theorem (the Principal Theorem) of wide generality, and then examples are given of some classes of varieties satisfying the conditions of the theorem. In Part II we find the base for the prime ideals associated with Veronesean varieties and varieties of Segre. The latter are particularly interesting since they represent (1, 1), without exception, the points of a multiply-projective space.

scholarly journals GENERATORS FOR $\mathcal{H}$-INVARIANT PRIME IDEALS IN $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$

2004 ◽  
Vol 47 (1) ◽  
pp. 163-190 ◽  
Author(s):  
Stéphane Launois
Keyword(s):  
Prime Ideal ◽  
Subject Classification ◽  
Prime Ideals ◽  
Generating Set ◽  
Maximal Orders ◽  
The Matrix ◽  
Secondary 16W35 ◽  
The Ideal

AbstractIt is known that, for generic $q$, the $\mathcal{H}$-invariant prime ideals in $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$ are generated by quantum minors (see S. Launois, Les idéaux premiers invariants de $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$, J. Alg., in press). In this paper, $m$ and $p$ being given, we construct an algorithm which computes a generating set of quantum minors for each $\mathcal{H}$-invariant prime ideal in $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$. We also describe, in the general case, an explicit generating set of quantum minors for some particular $\mathcal{H}$-invariant prime ideals in $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$. In particular, if $(Y_{i,\alpha})_{(i,\alpha)\in[[1,m]]\times[[1,p]]}$ denotes the matrix of the canonical generators of $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$, we prove that, if $u\geq3$, the ideal in $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$ generated by $Y_{1,p}$ and the $u\times u$ quantum minors is prime. This result allows Lenagan and Rigal to show that the quantum determinantal factor rings of $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$ are maximal orders (see T. H. Lenagan and L. Rigal, Proc. Edinb. Math. Soc.46 (2003), 513–529).AMS 2000 Mathematics subject classification: Primary 16P40. Secondary 16W35; 20G42

Reflections in extended Bianchi groups

1994 ◽  
Vol 115 (1) ◽  
pp. 13-25 ◽  
Author(s):  
L. Ya. Vulakh
Keyword(s):  
Complex Plane ◽  
Finite Index ◽  
Class Group ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Ring Of Integers ◽  
Linear Fractional Transformations ◽  
The Ideal

Letdbe a positive square-free number. LetOdbe the ring of integers in Q(√ −d). The groupsPSL(2, Od) are called collectively the Bianchi groups. The extended Bianchi groupBdis the maximal discrete extension ofPSL(2,Od) inPSL(2, C). The groupBdacts by linear fractional transformations on the complex plane C. LetRBdbeBdwith the generator θ,, adjoined. (RBdwill be also called an extended Bianchi group (cf. [18])). A group is said to be reflective if it contains a Coxeter subgroup (i.e. a subgroup generated by reflections) of finite index. The groupsRBdand their subgroups have been investigated in [3, 6, 13, 15, 16, 17, 18, 20, 21]. In 1892 Bianchi[3] proved thatPGL(2,Od)⋊{θ} is reflective ifd≤ 19,d╪ 14 or 17. Vinberg [18] proved that if the groupRBdis reflective, then the orders of all elements of the ideal-class group of the field Q(√ −d) should divide 4. Shaiheev [16] proved that there are only finitely many reflective extended Bianchi groups and found all of them ford≤ 30. Similar results are obtained for groupsPGL(2,Od) ⋊ {θ} which, as Shvartsman [17] showed, are reflective only whend= 1, 2, 5, 6, 10, 13, 21, providedd= 1 or 2 (mod 4).

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