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 \(\overline F (X)\) 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 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.

让ķ是由不可约多项式定义多个字段˚F（X）∈ℤ[ X ]和ℤ _ķ其整数环。对于每一个素数p，我们给出充分必要条件˚F（X）是保证生存的确切[Rℤ的素理想_ķ躺在上面p，其中\（\划线F（X）\）因素的权力- [R摩尼\（{\ mathbb {F} _p} [X] \）中的不可约多项式。与SK Khanduja和M.Kumar（2010）给出的结果相比，给定的结果呈现出较弱的条件，这保证了精确存在r prime _K的主要理想位于p之上。我们进一步指定ℤ的每一个素理想_ķ躺在上面p，分枝的指数，残留程度，和p -发电机。