## Rocky Mountain Journal of Mathematics

### Non-monogenity in a family of octic fields

#### Abstract

Let $m$ be a square-free integer, $m\equiv 2,3\pmod 4$. We show that the number field $K=\mathbb{Q} (i,\sqrt [4]{m})$ is non-monogene, that is, it does not admit any power integral bases of type $\{1,\alpha ,\ldots ,\alpha ^7\}$. In this infinite parametric family of Galois octic fields we construct an integral basis and show non-monogenity using congruence considerations only.

Our method yields a new approach to consider monogenity or to prove non-monogenity in algebraic number fields which is applicable for parametric families of number fields. We calculate the index of elements as polynomials dependent upon the parameter, factor these polynomials, and consider systems of congruences according to the factors.

#### Article information

Source
Rocky Mountain J. Math., Volume 47, Number 3 (2017), 817-824.

Dates
First available in Project Euclid: 24 June 2017

https://projecteuclid.org/euclid.rmjm/1498269812

Digital Object Identifier
doi:10.1216/RMJ-2017-47-3-817

Mathematical Reviews number (MathSciNet)
MR3682150

Zentralblatt MATH identifier
1381.11102

#### Citation

Gaál, István; Remete, László. Non-monogenity in a family of octic fields. Rocky Mountain J. Math. 47 (2017), no. 3, 817--824. doi:10.1216/RMJ-2017-47-3-817. https://projecteuclid.org/euclid.rmjm/1498269812

#### References

• Y. Bilu, I. Gaál and K. Győry, Index form equations in sextic fields: A hard computation, Acta Arith. 115 (2004), 85–96.
• Mu-Ling Chang, Non-monogenity in a family of sextic fields, J. Numer. Th. 97 (2002), 252–268.
• B.W. Char, K.O. Geddes, G.H. Gonnet, B.L. Leong, M.B. Monagan and S.M. Watt, Maple V language reference manual, Springer, New York, 1991.
• John Paul Cook, Computing integral bases, http://math.ou.edu/~jcook/LaTeX/integralbases.pdf.
• T. Funakura, On integral bases of pure quartic fields, Math. J. Okayama Univ. 26 (1984), 27–41.
• I. Gaál, Power integral bases in composits of number fields, Canad. Math. Bull. 41 (1998), 158–161.
• I. Gaál, Solving index form equations in fields of degree nine with cubic subfields, J. Symb. Comp. 30 (2000), 181–193.
• ––––, Diophantine equations and power integral bases, Birkhäuser, Boston, 2002.
• I. Gaál and K. Győry, Index form equations in quintic fields, Acta Arith. 89 (1999), 379–396.
• I. Gaál, P. Olajos and M. Pohst, Power integral bases in orders of composits of number fields, Exp. Math. 11 (2002), 87–90.
• I. Gaál, A. Pethő and M. Pohst, Simultaneous representation of integers by a pair of ternary quadratic forms with an application to index form equations in quartic number fields, J. Numer. Th. 57 (1996), 90–104.
• I. Gaál and M. Pohst, Power integral bases in a parametric family of totally real cyclic quintics, Math. Comp. 66 (1997), 1689–1696.
• I. Gaál, L. Remete and T. Szabó, Calculating power integral bases by solving relative Thue equations, Tatra Math. Publ. 59 (2014), 79-–92.
• I. Gaál and N. Schulte, Computing all power integral bases of cubic number fields, Math. Comp. 53 (1989), 689–696.
• I. Gaál and T. Szabó, Power integral bases in parametric families of biquadratic fields, J. Alg. Numer. Th. App. 21 (2012), 105–114.
• A. Hameed, T. Nakahara, S.M. Husnine and S. Ahmad, On the existence of canonical number system in certain classes of pure algebraic number fields, J. Prime Res. Math. 7 (2011), 19–24.
• J.G. Huard, B.K. Spearman and K.S. Williams, Integral bases for quartic fields with quadratic subfields, J. Numer. Th. 51 (1995), 87–102.
• W. Narkiewicz, Elementary and analytic theory of algebraic numbers, Second edition, Springer, New York, 1974.
• P. Olajos, Power integral bases in orders of composite fields II, Ann. Univ. Sci. Budapest 46 (2003), 35–41.