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Prime ideal factorization in a number field via Newton polygons. (English) Zbl 07361083

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 [Manuscr. Math. 131, No. 3–4, 323–334 (2010; Zbl 1216.12007)], 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.

MSC:

11Y05 Factorization
11Y40 Algebraic number theory computations
11S05 Polynomials

Citations:

Zbl 1216.12007
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Full Text: DOI

References:

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