On power basis of a class of algebraic number fields

2016 ◽  
Vol 12 (08) ◽  
pp. 2317-2321 ◽  
Author(s):  
Bablesh Jhorar ◽  
Sudesh K. Khanduja
Keyword(s):  
Polynomial Ring ◽  
Algebraic Number ◽  
Minimal Polynomial ◽  
Sufficient Conditions ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Rational Numbers ◽  
Integral Basis ◽  
Algebraic Number Fields ◽  
Necessary And Sufficient

Let [Formula: see text] be an algebraic number field with [Formula: see text] in the ring [Formula: see text] of algebraic integers of [Formula: see text] and [Formula: see text] be the minimal polynomial of [Formula: see text] over the field [Formula: see text] of rational numbers. In 1977, Uchida proved that [Formula: see text] if and only if [Formula: see text] does not belong to [Formula: see text] for any maximal ideal [Formula: see text] of the polynomial ring [Formula: see text] (see [Osaka J. Math. 14 (1977) 155–157]). In this paper, we apply the above result to obtain some necessary and sufficient conditions involving the coefficients of [Formula: see text] for [Formula: see text] to equal [Formula: see text] when [Formula: see text] is a trinomial of the type [Formula: see text]. In the particular case when [Formula: see text], it is deduced that [Formula: see text] is an integral basis of [Formula: see text] if and only if either (i) [Formula: see text] and [Formula: see text] or (ii) [Formula: see text] divides [Formula: see text] and [Formula: see text].

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 Computing subfields in algebraic number fields

1990 ◽  
Vol 49 (3) ◽  
pp. 434-448 ◽  
Author(s):  
John D. Dixon
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Algebraic Number ◽  
Minimal Polynomial ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Local Information ◽  
Algebraic Number Fields

AbstractLet K:= Q(α) be an algebraic number field which is given by specifying the minimal polynomial f(X) for α over Q. We describe a procedure for finding the subfields L of K by constructing pairs (w(X), g(X)) of polynomials over Q such that L= Q(w(α)) and g(X) is the minimal polynomial for w(α). The construction uses local information obtained from the Frobenius-Chebotarev theorem about the Galois group Gal(f), and computations over p-adic extensions of Q.

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.

Characterization of primes dividing the index of a trinomial

2017 ◽  
Vol 13 (10) ◽  
pp. 2505-2514 ◽  
Author(s):  
Anuj Jakhar ◽  
Sudesh K. Khanduja ◽  
Neeraj Sangwan
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Sufficient Conditions ◽  
Algebraic Number Field ◽  
Necessary And Sufficient Conditions ◽  
Algebraic Integers ◽  
Ring Of Algebraic Integers ◽  
Necessary And Sufficient

Let [Formula: see text] denote the ring of algebraic integers of an algebraic number field [Formula: see text], where [Formula: see text] is a root of an irreducible trinomial [Formula: see text] belonging to [Formula: see text]. In this paper, we give necessary and sufficient conditions involving only [Formula: see text] for a given prime [Formula: see text] to divide the index of the subgroup [Formula: see text] in [Formula: see text]. In particular, we deduce necessary and sufficient conditions for [Formula: see text] to be equal to [Formula: see text].

ON A THEOREM OF DEDEKIND

2008 ◽  
Vol 04 (06) ◽  
pp. 1019-1025 ◽  
Author(s):  
SUDESH K. KHANDUJA ◽  
MUNISH KUMAR
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Minimal Polynomial ◽  
Algebraic Number Field ◽  
Rational Numbers ◽  
Prime Ideals ◽  
Algebraic Integers ◽  
Dedekind Domains ◽  
Modulo P ◽  
Polynomial Formula

Let K = ℚ(θ) be an algebraic number field with θ in the ring AK of algebraic integers of K and f(x) be the minimal polynomial of θ over the field ℚ of rational numbers. For a rational prime p, let [Formula: see text] be the factorization of the polynomial [Formula: see text] obtained by replacing each coefficient of f(x) modulo p into product of powers of distinct monic irreducible polynomials over ℤ/pℤ. Dedekind proved that if p does not divide [AK : ℤ[θ]], then the factorization of pAK as a product of powers of distinct prime ideals is given by [Formula: see text], with 𝔭i = pAK + gi(θ)AK, and residual degree [Formula: see text]. In this paper, we prove that if the factorization of a rational prime p in AK satisfies the above-mentioned three properties, then p does not divide [AK:ℤ[θ]]. Indeed the analogue of the converse is proved for general Dedekind domains. The method of proof leads to a generalization of one more result of Dedekind which characterizes all rational primes p dividing the index of K.

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.

On Lattices in a Module Over a Matrix Algebra

1966 ◽  
Vol 9 (1) ◽  
pp. 57-61 ◽  
Author(s):  
Nobuo Nobusawa
Keyword(s):  
Algebraic Number ◽  
Quadratic Forms ◽  
Matrix Algebra ◽  
Symmetric Matrix ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Algebraic Number Fields ◽  
Left Module ◽  
Bilinear Mapping ◽  
The Matrix

Let A be the matrix algebra of type n × n over a finite algebraic number field F, and V the module of matrices of type n × m over F. V is naturally an A-left module. Given a non-singular symmetric matrix S of type m × m over F, we have a bilinear mapping f of V on A such that f(x, y) = xSy' for elements x and y in V where y' is the transpose of y. In this case, corresponding to the arithmetic of A([l]), the arithmetical theory of V will be discussed to some extent as we establish the arithmetic of quadratic forms over algebraic number fields ([2]). In this note, we shall define a lattice in V with respect to a maximal order in A. and determine its structure (Theorem 1), and after giving a structure of a complement of a lattice (Theorem 2), we shall give a finiteness theorem of class numbers of lattices under some assumption (Theorem 3).

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.

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