Prolongations of valuations to finite extensions

Abstract

Let ${K=\mathbb{Q}(\theta)}$ be an algebraic number field with θ in the ring A _K of algebraic integers of K and f(x) be the minimal polynomial of θ over the field ${\mathbb{Q}}$ of rational numbers. For a rational prime p, let ${\bar{f}(x)\,=\,\bar{g}_{1}(x)^{e_{1}}....\bar{g}_{r}(x)^{e_{r}}}$ be the factorization of the polynomial ${\bar{f}(x)}$ obtained by reducing coefficients of f(x) modulo p into a product of powers of distinct irreducible polynomials over ${\mathbb{Z}/p\mathbb{Z}}$ with g _i (x) monic. Dedekind proved that if p does not divide [${A_{K}:\mathbb{Z}}$ [θ]], then ${pA_{K}=\wp_{1}^{e_{1}}\ldots\wp_{r}^{e_{r}}}$, where ${\wp_{1},\ldots,\wp_{r}}$ are distinct prime ideals of A _K , ${\wp_{i}=pA_{K}+g_{i}(\theta)A_{K}}$ having residual degree equal to the degree of ${\bar{g}_{i}(x)}$. He also proved that p does not divide [${A_{K}:\mathbb{Z}}$[θ]] if and only if for each i, either e _i  = 1 or ${\bar{g}_{i}(x)}$ does not divide ${\bar{M}(x)}$ where ${M(x)=\frac{1}{p}(f(x)-g_{1}(x)^{e_{1}}....g_{r}(x)^{e_{r}})}$. Our aim is to give a weaker condition than the one given by Dedekind which ensures that if the polynomial ${\bar{f}(x)}$ factors as above over ${\mathbb{Z}/p\mathbb{Z}}$, then there are exactly r prime ideals of A _K lying over p, with respective residual degrees ${\deg \bar {g}_{1}(x),...,\deg \bar {g}_{r}(x)}$ and ramification indices e ₁, ..., e _r . 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.