Prime Ideal Factorization in a Number Field via Newton Polygons

Let K be a number field defined by an irreducible polynomial F ( X ) ∈ ℤ[ X ] and 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 ℤ K lying above p , where F ¯ ( X ) factors into po...

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Bibliographic Details
Published in:Czechoslovak mathematical journal 2021-06, Vol.71 (2), p.529-543
Main Author: El Fadil, Lhoussain
Format: Article
Language:English
Subjects:
Analysis
Convex and Discrete Geometry
Mathematical Modeling and Industrial Mathematics
Mathematics
Mathematics and Statistics
Number theory
Ordinary Differential Equations
Polynomials
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Summary:Let K be a number field defined by an irreducible polynomial F ( X ) ∈ ℤ[ X ] and 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 ℤ K lying above p , where F ¯ ( X ) factors into powers of r monic irreducible polynomials in 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 ℤ K lying above p . We further specify for every prime ideal of ℤ K lying above p , the ramification index, the residue degree, and a p -generator.
ISSN:0011-4642
1572-9141
DOI:10.21136/CMJ.2021.0516-19