Corps biquadratiques monogènes

“…This section is devoted to the results due to Gras and Tanoè [15]. We start with a result due to Williams [22] on the integral basis of K. In view of our canonic form of K we state William's result in the following form:…”

Section: Maximal Orders and Power Integral Basesmentioning

confidence: 98%

On biquadratic fields that admit unit power integral basis

Pethö

Ziegler

2011

Acta Math Hung

View full text Add to dashboard Cite

show abstract

“…R e m a r k. M.-N. Gras and F. Tanoé gave a necessary and sufficient condition that the ring of integers in a biquadratic field K = Q( √ mn, √ dn) has a power basis, i.e., K is monogenic as follows [3]; Vol. 83, 2004 Power integral bases in algebraic number fields 311…”

Section: Propositionmentioning

confidence: 99%

“…Further M.-N. Gras and F. Tanoé gave a necessary and sufficient condition for monogenesis of biquadratic fields by using a diophantine equation of degree 4 [3], and a new family of infinitely many monogenic biquadratic fields was constructed by the first author [4].…”

Section: Introductionmentioning

confidence: 99%

Power integral bases in algebraic number fields whose Galois groups are 2-elementary abelian

Motoda

Nakahara

2004

Arch. Math.

View full text Add to dashboard Cite

T o t h e m e m o r y o f P r o f e s s o r D r. E r i c h L a m p r e c h t Abstract. Let K be a biquadratic field. M.-N. Gras and F. Tanoé gave a necessary and sufficient condition that K is monogenic by using a diophantine equation of degree 4 [3]. We consider algebraic extension fields of higher degree. Let F be a Galois extension field over the rationals Q whose Galois group is 2-elementary abelian. Then we shall prove that F of degree [F : Q] 8, is monogenic if and only if F = Q( √ −1, √ −2, √ −3) = Q(ζ 24 ) under a suitable condition for the case of degree 8.

show abstract

“…These fields were considered several authors. M. N. Gras, and F. Tanoe ( [9]) have found necessary and sufficient conditions for biquadratic fields being monogenic. I. Gaál, A. Pethő and M. Pohst ( [7]) gave an algorithm for determining the minimal index and all elements with minimal index in the totally real case using the integral basis described by K. S. Williams ([14]).…”

Section: Introductionmentioning

confidence: 99%

On elements with index of the form 2a3b in a parametric family of biquadratic fields

Jadrijević

2015

Glas. Mat. Ser. III

View full text Add to dashboard Cite

Abstract. In this paper we give some results about primitive integral elements α in the family of bicyclic biquadratic fields Lc = Q( (c − 2) c, (c + 4) c) which have index of the form µ (α) = 2 a 3 b and coprime coordinates in given integral bases. Precisely, we show that if c ≥ 11 and α is an element with index µ (α) = 2 a 3 b ≤ c + 1, then α is an element with minimal index µ (α) = µ (Lc) = 12. We also show that for every integer C 0 ≥ 3 we can find effectively computable constants M 0 (C 0 ) and N 0 (C 0 ) such that if c ≤ C 0 , than there are no elements α with index of the form µ (α) = 2 a 3 b , where a > M (C 0 ) or b > N (C 0 ).

show abstract

Corps biquadratiques monogènes

Cited by 21 publications

References 10 publications

On biquadratic fields that admit unit power integral basis

On biquadratic fields that admit unit power integral basis

Power integral bases in algebraic number fields whose Galois groups are 2-elementary abelian

On elements with index of the form 2a3b in a parametric family of biquadratic fields

Contact Info

Product

Resources

About