Previous |  Up |  Next

Article

Full entry | Fulltext not available (moving wall 24 months)      Feedback
Keywords:
prime factorization; valuation; $\phi $-expansion; Newton polygon
Summary:
Let $K$ be a number field defined by an irreducible polynomial $F(X)\in \mathbb Z[X]$ and $\mathbb Z_K$ its ring of integers. For every prime integer $p$, we give sufficient and necessary conditions on $F(X)$ that guarantee the existence of exactly $r$ prime ideals of $\mathbb Z_K$ lying above $p$, where $\bar {F}(X)$ factors into powers of $r$ monic irreducible polynomials in $\mathbb F_p[X]$. The given result presents a weaker condition than that given by S. K. Khanduja and M. Kumar (2010), which guarantees the existence of exactly $r$ prime ideals of $\mathbb Z_K$ lying above $p$. We further specify for every prime ideal of $\mathbb Z_K$ lying above $p$, the ramification index, the residue degree, and a $p$-generator.
References:
[1] Bauer, M.: Über die ausserwesentliche Diskriminantenteiler einer Gattung. Math. Ann. 64 (1907), 573-576 German \99999JFM99999 38.0249.02. DOI 10.1007/BF01450065 | MR 1511458
[2] Bauer, M.: Zur allgemeinen Theorie der algebraischen Grösen. J. Reine Angew. Math. 132 (1907), 21-32 German. DOI 10.1515/crll.1907.132.21 | MR 1580710
[3] Cohen, S. D., Movahhedi, A., Salinier, A.: Factorization over local fields and the irreducibility of generalized difference polynomials. Mathematika 47 (2000), 173-196. DOI 10.1112/S0025579300015801 | MR 1924496 | Zbl 1018.12001
[4] Dedekind, R.: Über den Zusammenhang zwischen der Theorie der Ideale und der Theorie der höheren Congruenzen. Abh. Math. Klasse Königlichen Gesellsch. Wiss. Göttingen 23 (1878), 3-37 German.
[5] Fadil, L. El, Montes, J., Nart, E.: Newton polygons and $p$-integral bases of quartic number fields. J. Algebra Appl. 11 (2012), Article ID 1250073, 33 pages. DOI 10.1142/S0219498812500739 | MR 2959422 | Zbl 1297.11134
[6] Guàrdia, J., Montes, J., Nart, E.: Newton polygons of higher order in algebraic number theory. Trans. Am. Math. Soc. 364 (2012), 361-416. DOI 10.1090/S0002-9947-2011-05442-5 | MR 2833586 | Zbl 1252.11091
[7] Hensel, K.: Untersuchung der Fundamentalgleichung einer Gattung für eine reelle Primzahl als Modul und Bestimmung der Theiler ihrer Discriminante. J. Reine Angew. Math. 113 (1894), 61-83 German \99999JFM99999 25.0135.03. DOI 10.1515/crll.1894.113.61 | MR 1580345
[8] Khanduja, S. K., Kumar, M.: Prolongations of valuations to finite extensions. Manuscr. Math. 131 (2010), 323-334. DOI 10.1007/s00229-009-0320-1 | MR 2592083 | Zbl 1216.12007
[9] MacLane, S.: A construction for absolute values in polynomial rings. Trans. Am. Math. Soc. 40 (1936), 363-395 \99999JFM99999 62.1106.02. DOI 10.1090/S0002-9947-1936-1501879-8 | MR 1501879
Partner of
EuDML logo