TY  - JOUR
AU  - Gaál, István
TI  - Calculating generators of power integral bases in pure sextic fields
JF  - FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI
J2  - FUNCT APPROX COMMENT MATH
VL  - 70
PY  - 2024
IS  - 1
SP  - 85
EP  - 100
PG  - 16
SN  - 0208-6573
DO  - 10.7169/facm/2111
UR  - https://m2.mtmt.hu/api/publication/34788978
ID  - 34788978
LA  - English
DB  - MTMT
ER  - 

TY  - GEN
AU  - Lhoussain, El Fadil
AU  - Gaál, István
TI  - On the monogenity of quartic number fields defined by $x<SUP>4</SUP>+ax<SUP>2</SUP>+b$
PY  - 2024
PG  - 23
UR  - https://m2.mtmt.hu/api/publication/34788959
ID  - 34788959
LA  - English
DB  - MTMT
ER  - 

TY  - JOUR
AU  - Gaál, István
AU  - Remete, László
TI  - On the monogenity of pure quartic relative extensions of Q(i)
JF  - ACTA SCIENTIARUM MATHEMATICARUM (SZEGED)
J2  - ACTA SCI MATH (SZEGED)
VL  - 89
PY  - 2023
IS  - 3-4
SP  - 357
EP  - 371
PG  - 15
SN  - 0001-6969
DO  - 10.1007/s44146-023-00092-9
UR  - https://m2.mtmt.hu/api/publication/34039284
ID  - 34039284
AB  - We consider pure quartic relative extensions of the number field {{\mathbb {Q}}}(i) Q ( i ) of type K={{\mathbb {Q}}}(\root 4 \of {a+bi}) K = Q ( a + b i 4 ) , where a,b\in {{\mathbb {Z}}} a , b ∈ Z and b\ne 0 b ≠ 0 , such that a+bi\in {{\mathbb {Z}}}[i] a + b i ∈ Z [ i ] is square-free. We describe integral bases of these fields. The index form equation is reduced to a relative cubic Thue equation over {{\mathbb {Q}}}(i) Q ( i ) and some corresponding quadratic form equations. We consider monogenity of K and relative monogenity of K over {{\mathbb {Q}}}(i) Q ( i ) . We shall show how our former method based on the factors of the index form can be used in the relative case to exclude relative monogenity in some cases.
LA  - English
DB  - MTMT
ER  - 

TY  - JOUR
AU  - Gaál, István
TI  - Monogenity in totally real extensions of imaginary quadratic fields with an application to simplest quartic fields
JF  - ACTA SCIENTIARUM MATHEMATICARUM (SZEGED)
J2  - ACTA SCI MATH (SZEGED)
VL  - 89
PY  - 2023
IS  - 1-2
SP  - 3
EP  - 12
PG  - 10
SN  - 0001-6969
DO  - 10.1007/s44146-023-00081-y
UR  - https://m2.mtmt.hu/api/publication/33902457
ID  - 33902457
AB  - We describe an efficient algorithm to calculate generators of power integral bases in composites of totally real fields and imaginary quadratic fields with coprime discriminants. We show that the calculation can be reduced to solving index form equations in the original totally real fields. We illustrate our method by investigating monogenity in the infinite parametric family of imaginary quadratic extensions of the simplest quartic fields.
LA  - English
DB  - MTMT
ER  - 

TY  - JOUR
AU  - Gaál, István
TI  - On the monogenity of totally complex pure sextic fields
JF  - JP JOURNAL OF ALGEBRA, NUMBER THEORY AND APPLICATIONS
J2  - JP J ALGEBR NUM THEOR APPL
VL  - 60
PY  - 2023
IS  - 2
SP  - 85
EP  - 96
PG  - 12
SN  - 0972-5555
DO  - 10.17654/0972555523006
UR  - https://m2.mtmt.hu/api/publication/33885644
ID  - 33885644
AB  - Let 0 not equal m is an element of Z and alpha = (6)root m. According to the results of El Fadil [3], alpha generates a power integral basis in K = Q(alpha) , if and only if m is square-free, m not equivalent to 1(mod 4) and m not equivalent to +/- 1 (mod 9).In this note, we consider the case m < 0 and present an efficient method to calculate generators of power integral bases in totally complex pure sextic fields. Using this method, we performed an extensive calculation for this type of fields for 0 > m > -5000. In these 1521 fields, we did not find any other (non-equivalent) generators of power integral bases with coefficients in absolute values < 10(100) in the basis {1, alpha, alpha(2) , alpha(3) , alpha(4) , alpha(5)}.
LA  - English
DB  - MTMT
ER  - 

TY  - JOUR
AU  - El Fadil, Lhoussain
AU  - Gaál, István
TI  - Integral Bases and Monogenity of Pure Number Fields with Non-Square Free Parameters up to Degree 9
JF  - TATRA MOUNTAINS MATHEMATICAL PUBLICATIONS
J2  - TATRA MT MATH PUBL
VL  - 83
PY  - 2023
IS  - 1
SP  - 61
EP  - 86
PG  - 26
SN  - 1210-3195
DO  - 10.2478/tmmp-2023-0006
UR  - https://m2.mtmt.hu/api/publication/33778162
ID  - 33778162
AB  - Let K be a pure number field generated by a root α of a monic irreducible polynomial f ( x )= x n − m with m a rational integer and 3 ≤ n ≤ 9 an integer. In this paper, we calculate an integral basis of ℤ K , and we study the monogenity of K , extending former results to the case when m is not necessarily square-free. Collecting and completing the corresponding results in this more general case, our purpose is to provide a parallel to [Gaál, I.—Remete, L.: Power integral bases and monogenity of pure fields ,J.Number Theory, 173 (2017), 129–146], where only square-free values of m were considered.
LA  - English
DB  - MTMT
ER  - 

TY  - GEN
AU  - LHOUSSAIN, EL FADIL
AU  - Gaál, István
TI  - On integral bases and monogenity of pure octic number fields with non-square free parameters
PY  - 2022
UR  - https://m2.mtmt.hu/api/publication/33880787
ID  - 33880787
LA  - English
DB  - MTMT
ER  - 

TY  - JOUR
AU  - Gaál, István
TI  - On the monogenity of certain binomial compositions
JF  - JP JOURNAL OF ALGEBRA, NUMBER THEORY AND APPLICATIONS
J2  - JP J ALGEBR NUM THEOR APPL
VL  - 57
PY  - 2022
SP  - 1
EP  - 16
PG  - 16
SN  - 0972-5555
DO  - 10.17654/0972555522026
UR  - https://m2.mtmt.hu/api/publication/33590843
ID  - 33590843
LA  - English
DB  - MTMT
ER  - 

TY  - JOUR
AU  - Gaál, István
AU  - Pohst, Michael E.
TI  - On calculating the number N(D) of global cubic fields F of given discriminant D
JF  - JOURNAL OF NUMBER THEORY
J2  - J NUMBER THEORY
VL  - 236
PY  - 2022
SP  - 479
EP  - 491
PG  - 13
SN  - 0022-314X
DO  - 10.1016/j.jnt.2021.08.017
UR  - https://m2.mtmt.hu/api/publication/32799543
ID  - 32799543
LA  - English
DB  - MTMT
ER  - 

TY  - JOUR
AU  - Gaál, István
AU  - Pohst, Maximilian C.
AU  - Pohst, Michael E.
TI  - On computing integral points of a Mordell curve - the method of Wildanger revisited
JF  - EXPERIMENTAL MATHEMATICS
J2  - EXP MATH
VL  - 30
PY  - 2021
IS  - 1
SP  - 127
EP  - 134
PG  - 8
SN  - 1058-6458
DO  - 10.1080/10586458.2018.1502700
UR  - https://m2.mtmt.hu/api/publication/32684377
ID  - 32684377
AB  - We develop a new efficient algorithm for solving Mordell's equation. Our method is based on Wildanger's geometry of number approach. Major new ingredients come from Kummer theory and class field theory. This allows to enlarge the range of computations considerably. For explicit calculations, we used Magma and KANT.
LA  - English
DB  - MTMT
ER  -