| Title:
|
Ramification in quartic cyclic number fields $K$ generated by $x^4+px^2+p$ (English) |
| Author:
|
Pérez-Hernández, Julio |
| Author:
|
Pineda-Ruelas, Mario |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
146 |
| Issue:
|
4 |
| Year:
|
2021 |
| Pages:
|
471-481 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
ramification |
| Keyword:
|
cyclic quartic field |
| Keyword:
|
discriminant |
| Keyword:
|
index |
| MSC:
|
11R16 |
| MSC:
|
11S15 |
| idZBL:
|
Zbl 07442514 |
| idMR:
|
MR4336551 |
| DOI:
|
10.21136/MB.2021.0131-19 |
| . |
| Date available:
|
2021-11-08T16:21:35Z |
| Last updated:
|
2021-12-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149261 |
| . |
| Reference:
|
[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. Zbl 1080.11079, MR 2108046 |
| Reference:
|
[2] Alaca, Ş., Williams, K. S.: Introductory Algebraic Number Theory.Cambridge University Press, Cambridge (2004). Zbl 1035.11001, MR 2031707, 10.1017/cbo9780511791260 |
| Reference:
|
[3] Cohen, H.: A Course in Computational Algebraic Number Theory.Graduate Texts in Mathematics 138. Springer, Berlin (1993). Zbl 0786.11071, MR 1228206, 10.1007/978-3-662-02945-9 |
| Reference:
|
[4] Conrad, K.: Factoring after Dedekind.Available at https://kconrad.math.uconn.edu/blurbs/gradnumthy/dedekindf.pdf 7 pages. |
| Reference:
|
[5] Driver, E., Leonard, P. A., Williams, K. S.: Irreducible quartic polynomials with factorizations modulo $p$.Am. Math. Mon. 112 (2005), 876-890. Zbl 1211.11037, MR 2186831, 10.2307/30037628 |
| Reference:
|
[6] Engstrom, H. T.: On the common index divisors of an algebraic field.Trans. Am. Math. Soc. 32 (1930), 223-237 \99999JFM99999 56.0885.04. MR 1501535, 10.1090/S0002-9947-1930-1501535-0 |
| Reference:
|
[7] Guàrdia, J., Montes, J., Nart, E.: Higher Newton polygons in the computation of discriminants and prime ideals decomposition in number fields.J. Théor. Nombres Bordx. 23 (2011), 667-696. Zbl 1266.11131, MR 2861080, 10.5802/jtnb.782 |
| Reference:
|
[8] Guàrdia, J., Montes, J., Nart, E.: Newton polygons of higher order in algebraic number theory.Trans. Am. Math. Soc. 364 (2012), 361-416. Zbl 1252.11091, MR 2833586, 10.1090/S0002-9947-2011-05442-5 |
| Reference:
|
[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). Zbl 0615.12009 |
| Reference:
|
[10] Hudson, R. H., Williams, K. S.: The integers of a cyclic quartic field.Rocky Mt. J. Math. 20 (1990), 145-150. Zbl 0707.11078, MR 1057983, 10.1216/rmjm/1181073167 |
| Reference:
|
[11] Kappe, L.-C., Warren, B.: An elementary test for the Galois group of a quartic polynomial.Am. Math. Mon. 96 (1989), 133-137. Zbl 0702.11075, MR 0992075, 10.2307/2323198 |
| Reference:
|
[12] Llorente, P., Nart, E.: Effective determination of the decomposition of the rational primes in a cubic field.Proc. Am. Math. Soc. 87 (1983), 579-585. Zbl 0514.12003, MR 0687621, 10.1090/S0002-9939-1983-0687621-6 |
| Reference:
|
[13] Montes, J.: Polígonos de Newton de orden superior y aplicaciones aritméticas: Dissertation Ph.D.Universitat de Barcelona, Barcelona (1999), Spanish. |
| Reference:
|
[14] Spearman, B. K., Williams, K. S.: The index of a cyclic quartic field.Monatsh. Math. 140 (2003), 19-70. Zbl 1049.11111, MR 2007140, 10.1007/s00605-002-0547-3 |
| . |
