Geodesic
Ramification in quartic cyclic number fields $K$ generated by $x^4+px^2+p$
Mathematica Bohemica, Tome 146 (2021) no. 4, pp. 471-481
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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.
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.
DOI : 10.21136/MB.2021.0131-19
Classification : 11R16, 11S15
Keywords: ramification; cyclic quartic field; discriminant; index 
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Pérez-Hernández, Julio; Pineda-Ruelas, Mario. Ramification in quartic cyclic number fields $K$ generated by $x^4+px^2+p$. Mathematica Bohemica, Tome 146 (2021) no. 4, pp. 471-481. doi: 10.21136/MB.2021.0131-19

[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. | MR | JFM

[2] Alaca, Ş., Williams, K. S.: Introductory Algebraic Number Theory. Cambridge University Press, Cambridge (2004). | DOI | MR | JFM

[3] Cohen, H.: A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics 138. Springer, Berlin (1993). | DOI | MR | JFM

[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.: Irreducible quartic polynomials with factorizations modulo $p$. Am. Math. Mon. 112 (2005), 876-890. | DOI | MR | JFM

[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. | DOI | MR

[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. | DOI | MR | JFM

[8] Guàrdia, J., Montes, J., Nart, E.: Newton polygons of higher order in algebraic number theory. Trans. Am. Math. Soc. 364 (2012), 361-416. | DOI | MR | JFM

[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). | JFM

[10] Hudson, R. H., Williams, K. S.: The integers of a cyclic quartic field. Rocky Mt. J. Math. 20 (1990), 145-150. | DOI | MR | JFM

[11] Kappe, L.-C., Warren, B.: An elementary test for the Galois group of a quartic polynomial. Am. Math. Mon. 96 (1989), 133-137. | DOI | MR | JFM

[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. | DOI | MR | JFM

[13] Montes, J.: Polígonos de Newton de orden superior y aplicaciones aritméticas: Dissertation Ph.D. Universitat de Barcelona, Barcelona (1999), Spanish.

[14] Spearman, B. K., Williams, K. S.: The index of a cyclic quartic field. Monatsh. Math. 140 (2003), 19-70. | DOI | MR | JFM

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