Power integral basis for relative extensions of p^{n}-power Number Fields
Resumen
Let $L = K(\alpha)$ be an extension of the number field K, where $\alpha$ satisfies the monic irreducible polynomial $f(x)=x^{p^n}-\beta$ of power-prime degree belonging to ${\mathfrak{o}}_{K}[x]$ and $\mathfrak{o}_K$ is the integral closure of $K$.
The purpose of this paper is to study the monogenity of $L/K$ by using a new version of Dedekind's criterion. So, we give an integral basis of a family of number field of degree $2 p^n$ for some positive integer n. As an illustration, we get a slightly simpler computation of relative discriminant $D_{L/K }$.
Descargas
Citas
O. Boughaleb, A. Soullami and M. Sahmoudi, On relative monogeneity of a family of number fields defined by Xpn + aXps − b, Bol. Soc. Mat. Mex. 29 (2023).
M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, Massachusetts, 1969.
M. E Charkani and M. Sahmoudi, Sextic Extension with cubic subfield, JP Journal of Algebra, Number Theory et Applications, 34, no. 2, 139-150, (2014).
R. Dedekind, Uber den zussamenhang zwischen der theorie der ideals und der theorie der hoheren cyclotimy index, Abh. Akad. Wiss. Gottingen, Math.-Phys. KL 23 1-23, (1878)
J. Didi, M. Sahmoudi and A. Chillali, A class of number fields without odd rational prime index divisors and applications, J. Algebra and Related Topics (2024) (In press)
A. Frolich and J. Taylor, Algebraic Number Theory, Combridge Studies in Advenced Mathematics, 27, Cambridge University Press, (1993).
I. Gaal, An experiment on the monogenity of a family of trinomial, JP Journal of Algebra Number Theory Appl. 51, no. 1, 97–111 (2021).
M. N. Gras, Non monogeneite de l’anneau des entiers des extensions cycliques de Q de degre premier l ≥ 5, J. Number Theory 23, no. 3, 347-353 (1986).
M. Haghighi, Relative integral basis for algebraic number fields, Int. J. Math. Math. Sci. 9 no. 1, 97-104 (1986).
B. Jhorar and S.K. Khanduja, On power basis of a class of algebraic number fields, I. J. Number Theory , 12 no. 8, 2317–2321 (2016).
M.J. Lavalee and B.K. Sperman, K.S. Williams,Lifting monogenic cubic fields to monogenic sextic fields, Kodai math. J. 34, 410-425, (2011).
W. Narkizewicz, Elementary and Analytic Theory of Algebraic Numbers, Springer-Verlag, Second Edition, (1999).
J. Neukirch, Algebraic Number Theory, Springer Publication, (1999).
M. Sahmoudi, and M. E Charkani, On relative pure cyclic fields with power integral bases, Mathematica Bohemica, 1-12 148 (2022).
M. Sahmoudi and A. Soullami, On monogenicity of relative cubic-power estensions, Advances in Mathematics: Scientific Journal 9, no.9, 6817–6827 (2020).
M. Sahmoudi and A. Soullami, On Sextic Integral Bases Using Relative Quadratic Extention, Bol. Soc. Paran. Mat. 38 no.4, 175-180 (2020).
M. Sahmoudi, Explicit integral basis for a family of sextic field, Gulf J. Math., 4, 217-222 (2016).
A. Soullami, M. Sahmoudi and O. Bughaleb, On relative power integral bases in a family of Numbers fields, Rocky Mountain Journal of Mathematics, 51 no. 4, 1443-1452 (2021).
P. SCHMID, On criteria by Dedekind and Ore for integral ring extensions, Arch. Math., 84, 304-310 (2005).
B. K. Sperman and K.S. Williams, Relative integral bases for quartic fields over quadratic subfields, Acta Math. Hungar., 70, 185-192 (1996).
Derechos de autor 2025 Boletim da Sociedade Paranaense de Matemática
Esta obra está bajo licencia internacional Creative Commons Reconocimiento 4.0.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
