Minimal index of bicyclic biquadratic number fields. (English) Zbl 1445.11123

Arnóczki, Tímea; Nyul, Gábor

Rocky Mt. J. Math. 50, No. 1, 1-8 (2020).

Let \(K\) be an algebraic number field. For any primitive algebraic integer \(\alpha\in K\) the index \(I(\alpha)\) is defined as the module index of \(\mathbb Z[\alpha]\) in the ring of integers \(\mathbb Z_K\) of \(K\).
The minimal index of \(K\) is \[m_K=\min I(\alpha), \] the minimum taken for all primitive algebraic integers \(\alpha\) in \(K\). The field index of \(K\) is \[i_K=\gcd I(\alpha)\] the gcd taken for all primitive algebraic integers \(\alpha\) in \(K\).
There are only a few results describing the behaviour of minimal indices and field indices in certain types of algebraic number fields.
The authors investigate bicyclic biquadratic number fields of type \(\mathbb Q(\sqrt{m},\sqrt{n})\), where \(m\) and \(n\) are distinct square-free integers, both different from 0 and 1. The authors prove the following statements:
(1) Let \(a\in\{1,3\}\). Then, for any positive multiple \(A\) of \(a\), there exist infinitely many totally complex bicyclic biquadratic number fields with field index \(a\) and minimal index \(A\).
(2) Let \(a\in\{2,4,6,12\}\). Then, for any positive multiple \(A\) of \(2\)a, there exist infinitely many totally complex bicyclic biquadratic number fields with field index \(a\) and minimal index \(A\).
These theorems represent significant contributions to the existing literature on this topic.

Reviewer: István Gaál (Debrecen)

MSC:

11R16	Cubic and quartic extensions
11D57	Multiplicative and norm form equations

Keywords:

minimal index; field index; bicyclic biquadratic number field

References:

[1]	T. Arnóczki, “Elements with prime and small indices in bicyclic biquadratic number fields”, Period. Math. Hungar. 77:1 (2018), 83-93. \xoxMR3842972 · Zbl 1424.11153 · doi:10.1007/s10998-017-0233-9
[2]	R. Dedekind, “Über den Zusammenhang zwischen der Theorie der Ideale und der Theorie der höheren Congruenzen”, Abh. König. Ges. Wiss. Göttingen 23 (1878), 3-37.
[3]	J.-H. Evertse \andword K. Győry, Discriminant equations in Diophantine number theory, New Mathematical Monographs 32, Cambridge Univ. Press, 2017. \xoxMR3586280
[4]	I. Gaál, Diophantine equations and power integral bases: new computational methods, Birkhäuser, 2002. \xoxMR1896601 · Zbl 1016.11059
[5]	I. Gaál \andword B. Jadrijević, “Determining elements of minimal index in an infinite family of totally real bicyclic biquadratic number fields”, JP J. Algebra Number Theory Appl. 39:3 (2017), 307-326. · Zbl 1373.11077
[6]	I. Gaál, A. Pethö\serialcomma \andword M. Pohst, “On the indices of biquadratic number fields having Galois group \(V_4\)”, Arch. Math. \((\) Basel \() 57\):4 (1991), 357-361. \xoxMR1124498 · Zbl 0724.11049
[7]	I. Gaál, A. Pethö\serialcomma \andword M. Pohst, “On the resolution of index form equations in biquadratic number fields, III: The bicyclic biquadratic case”, J. Number Theory 53:1 (1995), 100-114. \xoxMR1344834 · Zbl 0853.11026
[8]	K. Győry, “Sur les polynômes à coefficients entiers et de discriminant donné, III”, Publ. Math. Debrecen 23:1-2 (1976), 141-165. \xoxMR437491 · Zbl 0354.10041
[9]	M. Hall, “Indices in cubic fields”, Bull. Amer. Math. Soc. 43:2 (1937), 104-108. \xoxMR1563496 · Zbl 0016.00803
[10]	B. Jadrijević, “Establishing the minimal index in a parametric family of bicyclic biquadratic fields”, Period. Math. Hungar. 58:2 (2009), 155-180. \xoxMR2531162 · Zbl 1265.11061
[11]	B. Jadrijević, “Solving index form equations in two parametric families of biquadratic fields”, Math. Commun. 14:2 (2009), 341-363. \xoxMR2743182 · Zbl 1257.11098
[12]	H. H. Kim \andword Z. Wolske, “Number fields with large minimal index containing quadratic subfields”, Int. J. Number Theory 14:9 (2018), 2333-2342. \xoxMR3855462 · Zbl 1437.11150
[13]	T. Nagel, “Zur Arithmetik der Polynome”, Abh. Math. Sem. Univ. Hamburg 1:1 (1922), 178-193. \xoxMR3069398 · JFM 48.0132.04
[14]	T. Nakahara, “On the indices and integral bases of non-cyclic but abelian biquadratic fields”, Arch. Math. \((\) Basel \() 41\):6 (1983), 504-508. \xoxMR731633 · Zbl 0513.12005
[15]	G. Nyul, “Power integral bases in totally complex biquadratic number fields”, Acta Acad. Paedagog. Agriensis Sect. Math. \((\) N.S.\() 28 (2001), 79-86\). \xoxMR1875563 · Zbl 0988.11011
[16]	G. Nyul, Monogenity of algebraic number fields, Ph.D. thesis, University of Debrecen, Hungary, 2007. In Hungarian.
[17]	B. K. Spearman, Q. Yang\serialcomma \andword J. Yoo, “Minimal indices of pure cubic fields”, Arch. Math. \((\) Basel \() 106\):1 (2016), 35-40. \xoxMR3451364 · Zbl 1333.11099
[18]	K. S. Williams, “Integers of biquadratic fields”, Canad. Math. Bull. 13 (1970), 519-526. \xoxMR279069 · Zbl 0205.35401