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Exponents of Prime Ideals in Number Fields

I have a question about “factoring out” common prime ideal factors in a number field.

Let $K$ be a number field with ring of integers $\mathcal{O}_K$. Let $\mathfrak{p}$ be a prime ideal of $\mathcal{O}_K$.

Now let $k \in K$. I know I can write $k = a/b$ for $a,b \in \mathcal{O}_K$. Now consider the fractional ideal $k\mathcal{O}_K$. Let $v_\mathfrak{p}(x)$ be the exponent that $\mathfrak{p}$ appears to in the prime factorization of this fractional ideal. We of course have that $v_\mathfrak{p}(x) = v_\mathfrak{p}(a)- v_\mathfrak{p}(b)$. Now if $v_\mathfrak{p}(x) \geq 0$, is it always possible to choose $a,b$ in such a way so that $v_\mathfrak{p}(b) = 0$. So in a way, are we able to “factor out” common factors in their prime decomposition?

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