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Prime Ideal Factorization in a Number Field via Newton Polygons

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Abstract

Let K be a number field defined by an irreducible polynomial F(X) ∈ ℤ[X] and ℤK 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 ℤK lying above p, where \(\overline F \left( X \right)\) factors into powers of r monic irreducible polynomials in \({\mathbb{F}_p}\left[ X \right]\). 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 ℤK lying above p. We further specify for every prime ideal of ℤK lying above p, the ramification index, the residue degree, and a p-generator.

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Acknowledgements

The author is deeply grateful to the anonymous referee, their valuable comments and suggestions have tremendously improved the quality of the paper. Also, he is very grateful to Professor Enric Nart for introducing him to Newton polygon’s techniques when he was a post-doc at CRM of Barcelona, Spain (2007–2008).

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Correspondence to Lhoussain El Fadil.

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El Fadil, L. Prime Ideal Factorization in a Number Field via Newton Polygons. Czech Math J 71, 529–543 (2021). https://doi.org/10.21136/CMJ.2021.0516-19

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