Prime ideal factorization in a number field via Newton polygons

El Fadil, Lhoussain. "Prime ideal factorization in a number field via Newton polygons." Czechoslovak Mathematical Journal 71.2 (2021): 529-543. <http://eudml.org/doc/297822>.

@article{ElFadil2021,
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.},
author = {El Fadil, Lhoussain},
journal = {Czechoslovak Mathematical Journal},
keywords = {prime factorization; valuation; $\phi $-expansion; Newton polygon},
language = {eng},
number = {2},
pages = {529-543},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Prime ideal factorization in a number field via Newton polygons},
url = {http://eudml.org/doc/297822},
volume = {71},
year = {2021},
}

TY - JOUR
AU - El Fadil, Lhoussain
TI - Prime ideal factorization in a number field via Newton polygons
JO - Czechoslovak Mathematical Journal
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 71
IS - 2
SP - 529
EP - 543
AB - 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.
LA - eng
KW - prime factorization; valuation; $\phi $-expansion; Newton polygon
UR - http://eudml.org/doc/297822
ER -