Czechoslovak Mathematical Journal, Vol. 71, No. 2, pp. 529-543, 2021
Prime ideal factorization in a number field via Newton polygons
Lhoussain El Fadil
Received December 2, 2019. Published online March 8, 2021.
Abstract: 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.
Keywords: prime factorization; valuation; $\phi$-expansion; Newton polygon
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Affiliations: Lhoussain El Fadil, Faculty of Sciences Dhar El Mahraz, Sidi Mohamed Ben Abdellah University, P.O. Box 1874 Atlas-Fes, Fès, Morocco, e-mail: lhouelfadil2@gmail.com
