Ramification in quartic cyclic number fields K generated by x 4 + p x 2 + p

Julio Pérez-Hernández; Mario Pineda-Ruelas

Mathematica Bohemica (2021)

  • Volume: 146, Issue: 4, page 471-481
  • ISSN: 0862-7959

Abstract

top
If K is the splitting field of the polynomial f ( x ) = x 4 + p x 2 + p and p is a rational prime of the form 4 + n 2 , we give appropriate generators of K to obtain the explicit factorization of the ideal q 𝒪 K , where q is a positive rational prime. For this, we calculate the index of these generators and integral basis of certain prime ideals.

How to cite

top

Pérez-Hernández, Julio, and Pineda-Ruelas, Mario. "Ramification in quartic cyclic number fields $K$ generated by $x^4+px^2+p$." Mathematica Bohemica 146.4 (2021): 471-481. <http://eudml.org/doc/298278>.

@article{Pérez2021,
abstract = {If $K$ is the splitting field of the polynomial $f(x)=x^4+px^2+p$ and $p$ is a rational prime of the form $4+n^2$, we give appropriate generators of $K$ to obtain the explicit factorization of the ideal $q\{\mathcal \{O\}\}_\{K\}$, where $q$ is a positive rational prime. For this, we calculate the index of these generators and integral basis of certain prime ideals.},
author = {Pérez-Hernández, Julio, Pineda-Ruelas, Mario},
journal = {Mathematica Bohemica},
keywords = {ramification; cyclic quartic field; discriminant; index},
language = {eng},
number = {4},
pages = {471-481},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Ramification in quartic cyclic number fields $K$ generated by $x^4+px^2+p$},
url = {http://eudml.org/doc/298278},
volume = {146},
year = {2021},
}

TY - JOUR
AU - Pérez-Hernández, Julio
AU - Pineda-Ruelas, Mario
TI - Ramification in quartic cyclic number fields $K$ generated by $x^4+px^2+p$
JO - Mathematica Bohemica
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 146
IS - 4
SP - 471
EP - 481
AB - If $K$ is the splitting field of the polynomial $f(x)=x^4+px^2+p$ and $p$ is a rational prime of the form $4+n^2$, we give appropriate generators of $K$ to obtain the explicit factorization of the ideal $q{\mathcal {O}}_{K}$, where $q$ is a positive rational prime. For this, we calculate the index of these generators and integral basis of certain prime ideals.
LA - eng
KW - ramification; cyclic quartic field; discriminant; index
UR - http://eudml.org/doc/298278
ER -

References

top
  1. Alaca, Ş., Spearman, B. K., Williams, K. S., The factorization of 2 in cubic fields with index 2, Far East J. Math. Sci. (FJMS) 14 (2004), 273-282. (2004) Zbl1080.11079MR2108046
  2. Alaca, Ş., Williams, K. S., 10.1017/cbo9780511791260, Cambridge University Press, Cambridge (2004). (2004) Zbl1035.11001MR2031707DOI10.1017/cbo9780511791260
  3. Cohen, H., 10.1007/978-3-662-02945-9, Graduate Texts in Mathematics 138. Springer, Berlin (1993). (1993) Zbl0786.11071MR1228206DOI10.1007/978-3-662-02945-9
  4. Conrad, K., Factoring after Dedekind, Available at https://kconrad.math.uconn.edu/blurbs/gradnumthy/dedekindf.pdf 7 pages. 
  5. Driver, E., Leonard, P. A., Williams, K. S., 10.2307/30037628, Am. Math. Mon. 112 (2005), 876-890. (2005) Zbl1211.11037MR2186831DOI10.2307/30037628
  6. Engstrom, H. T., 10.1090/S0002-9947-1930-1501535-0, Trans. Am. Math. Soc. 32 (1930), 223-237 9999JFM99999 56.0885.04. (1930) MR1501535DOI10.1090/S0002-9947-1930-1501535-0
  7. Guàrdia, J., Montes, J., Nart, E., 10.5802/jtnb.782, J. Théor. Nombres Bordx. 23 (2011), 667-696. (2011) Zbl1266.11131MR2861080DOI10.5802/jtnb.782
  8. Guàrdia, J., Montes, J., Nart, E., 10.1090/S0002-9947-2011-05442-5, Trans. Am. Math. Soc. 364 (2012), 361-416. (2012) Zbl1252.11091MR2833586DOI10.1090/S0002-9947-2011-05442-5
  9. Hardy, K., Hudson, R. H., Richman, D., Williams, K. S., Holtz, N. M., Calculation of the Class Numbers of Imaginary Cyclic Quartic Fields, Carleton-Ottawa Mathematical Lecture Note Series 7. Carleton University, Ottawa (1986). (1986) Zbl0615.12009
  10. Hudson, R. H., Williams, K. S., 10.1216/rmjm/1181073167, Rocky Mt. J. Math. 20 (1990), 145-150. (1990) Zbl0707.11078MR1057983DOI10.1216/rmjm/1181073167
  11. Kappe, L.-C., Warren, B., 10.2307/2323198, Am. Math. Mon. 96 (1989), 133-137. (1989) Zbl0702.11075MR0992075DOI10.2307/2323198
  12. Llorente, P., Nart, E., 10.1090/S0002-9939-1983-0687621-6, Proc. Am. Math. Soc. 87 (1983), 579-585. (1983) Zbl0514.12003MR0687621DOI10.1090/S0002-9939-1983-0687621-6
  13. Montes, J., Polígonos de Newton de orden superior y aplicaciones aritméticas: Dissertation Ph.D, Universitat de Barcelona, Barcelona (1999), Spanish. (1999) 
  14. Spearman, B. K., Williams, K. S., 10.1007/s00605-002-0547-3, Monatsh. Math. 140 (2003), 19-70. (2003) Zbl1049.11111MR2007140DOI10.1007/s00605-002-0547-3

NotesEmbed ?

top

You must be logged in to post comments.