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\begin{document}$${K=\mathbb{Q}(\theta)}$$\end{document} be an algebraic number field with θ in the ring AK of algebraic integers of K and f(x) be the minimal polynomial of θ over the field \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb{Q}}$$\end{document} of rational numbers. For a rational prime p, let \documentclass[12pt]{minimal}
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\begin{document}$${\bar{f}(x)\,=\,\bar{g}_{1}(x)^{e_{1}}....\bar{g}_{r}(x)^{e_{r}}}$$\end{document} be the factorization of the polynomial \documentclass[12pt]{minimal}
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\begin{document}$${\bar{f}(x)}$$\end{document} obtained by reducing coefficients of f(x) modulo p into a product of powers of distinct irreducible polynomials over \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb{Z}/p\mathbb{Z}}$$\end{document} with gi(x) monic. Dedekind proved that if p does not divide [\documentclass[12pt]{minimal}
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\begin{document}$${A_{K}:\mathbb{Z}}$$\end{document} [θ]], then \documentclass[12pt]{minimal}
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\begin{document}$${pA_{K}=\wp_{1}^{e_{1}}\ldots\wp_{r}^{e_{r}}}$$\end{document}, where \documentclass[12pt]{minimal}
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\begin{document}$${\wp_{1},\ldots,\wp_{r}}$$\end{document} are distinct prime ideals of AK, \documentclass[12pt]{minimal}
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\begin{document}$${\wp_{i}=pA_{K}+g_{i}(\theta)A_{K}}$$\end{document} having residual degree equal to the degree of \documentclass[12pt]{minimal}
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\begin{document}$${\bar{g}_{i}(x)}$$\end{document}. He also proved that p does not divide [\documentclass[12pt]{minimal}
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\begin{document}$${A_{K}:\mathbb{Z}}$$\end{document}[θ]] if and only if for each i, either ei = 1 or \documentclass[12pt]{minimal}
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\begin{document}$${\bar{g}_{i}(x)}$$\end{document} does not divide \documentclass[12pt]{minimal}
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\begin{document}$${\bar{M}(x)}$$\end{document} where \documentclass[12pt]{minimal}
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\begin{document}$${M(x)=\frac{1}{p}(f(x)-g_{1}(x)^{e_{1}}....g_{r}(x)^{e_{r}})}$$\end{document}. Our aim is to give a weaker condition than the one given by Dedekind which ensures that if the polynomial \documentclass[12pt]{minimal}
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\begin{document}$${\bar{f}(x)}$$\end{document} factors as above over \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb{Z}/p\mathbb{Z}}$$\end{document}, then there are exactly r prime ideals of AK lying over p, with respective residual degrees \documentclass[12pt]{minimal}
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\begin{document}$${\deg \bar {g}_{1}(x),...,\deg \bar {g}_{r}(x)}$$\end{document} and ramification indices e1, ..., er. In this paper, the above problem has been dealt with in a more general situation when the base field is a valued field (K, v) of arbitrary rank and K(θ) is any finite extension of K.
