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JP Journal of Algebra, Number Theory and Applications

JP Journal of Algebra, Number Theory and Applications
Volume 64, Issue 4, Pages 353 - 378 (July 2025)
http://dx.doi.org/10.17654/0972555525019

RAMIFICATION IN NUMBER FIELDS GENERATED BY WITH A RATIONAL PRIME

Julio Pérez-Hernández and Mario Pineda-Ruelas

Abstract:

If L is the splitting field of the polynomial and is a rational prime, then we have where D₈ is the dihedral group of order 8. We calculate the discriminant of L without knowing any integral basis of L; also, we obtain the explicit ramification of any rational prime q such that For this, we first study the ramification in the intermediate quartic field We give generators of each ramified rational prime. In some cases, we can make use of Dedekind’s Theorem; in other cases, we use the relative extension theory.

Keywords and phrases:

discriminant, relative discriminant, ramification, octic fields, index

Received: November 28, 2024; Accepted: March 7, 2025; Published: May 23, 2025

How to cite this article: Julio Pérez-Hernández and Mario Pineda-Ruelas, Ramification in number fields generated by with a rational prime, JP Journal of Algebra, Number Theory and Applications 64(4) (2025), 353-378. https://doi.org/10.17654/0972555525019

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1]S. Alaca, B. K. Spearman and K. Williams, The factorization of 2 in cubic fields with index 2, Far East Journal of Mathematical Sciences (FJMS) 14(3) (2004), 273-282.

[2]S. Alaca, B. K. Spearman and K. Williams, Explicit decomposition of a rational prime in a cubic field, International Journal of Mathematics and Mathematical Sciences (2005), 1-11.

[3]Conrad Keith, Factoring after Dedekind,

https://kconrad.math.uconn.edu/blurbs/gradnumthy/dedekindf.pdf.

[4]R. Dedekind, Theory of algebraic integers, Cambridge Mathematical Library, Translated from the 1877 French original and with an introduction by John Stillwell, 1997.

[5]I. Gaal, On the indices of biquadratic number fields having Galois group Arch. Math. 57 (1991), 357-361.

[6]J. G. Huard, B. K. Spearman and K. S. Williams, Integral Bases for Quartic Fields with Quadratic Subfields, Carleton University Centre for Research in Algebra and Number Theory Mathematical Research Series No. 4, 1991.

[7]K. Hardy, R. H. Hudson, D. Richman, K. S. Williams and N. M. Holtz, Calculation of the Class Numbers of Imaginary Cyclic Quartic Fields, Carleton-Ottawa Mathematical Lecture Note Series, No. 7, 1986.

[8]P. Llorente and E. Nart, Effective determination of the decomposition of the rational primes in a cubic field, Proceedings of the American Mathematical Society 87 (1983), 579-585.

[9]J. Guardia, J. Montes and E. Nart, Higher Newton polygons in the computation of discriminants and prime ideals decomposition in number fields, J. Théor. Nombres Bordx. 23 (2011), 667-696.

[10]J. Montes, Polígonos de Newton de orden superior y aplicaciones aritméticas, Dissertation, Universitat de Barcelona, 1999.

[11]J. Pérez-Hernández and M. Pineda-Ruelas, Ramification in quartic cyclic number fields K generated by Mathematica Bohemica 4 (2021), 471-481.

[12]P. Ribenboim, Classical Theory of Algebraic Numbers, Springer-Verlag, New York, 2001.

[13]B. K. Spearman and K. S. Williams, Relative integral bases for quartic fields over quadratic subfields, Acta Math. Hungar (1996), 185-192.

[14]B. K. Spearman and K. S. Williams, The Index of a Cyclic Quartic Field, Monatshefte für Mathematik 140 (2003), 19-70.

[15]K. S. Williams, Integers of biquadratic fields, Canadian Mathematical Bulletin 13 (1970), 519-526.

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