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Keywords:
prime factorization; valuation; $\phi $-expansion; Newton polygon
prime factorization; valuation; $\phi $-expansion; Newton polygon
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 (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.
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