Extension of valuation to the algebraic extension of a number field.

Question

I am trying to get the idea how we can extend the $p$-adic valuation on $\mathbb Q$ to an algebraic extension. In particular, how to extend the $p$-adic valuation for $p = 5$ from $\mathbb Q$ to $\mathbb Q(5^{1/3})$. Thank you.

Answer 1

The analogue of the $p$-adic valuation of a number field depends upon the prime ideals of its ring of integers. Let $K$ be a number field with ring of integers $\mathcal{O}_K$, and let $\mathfrak{p}\lhd\mathcal{O}_K$ be a non-zero prime. Then for any non-zero element $x\in K$, we can factor the fractional ideal $x\mathcal{O}_K\subset K$ uniquely into a product of prime ideals, say $x\mathcal{O}_K=\mathfrak{p}_1^{e_1}\mathfrak{p}_2^{e_2}\cdots$, where the $e_i\in\mathbb{Z}$ and each $\mathfrak{p}_i\lhd\mathcal{O}_K$ is prime. The $\mathfrak{p}_i$-adic valuation $\text{ord}_{\mathfrak{p}_i}(x)$ is then the exponent $e_i$ in the prime factorisation of $x\mathcal{O}_K$.

In your particular example, $K=\mathbb{Q}(\sqrt[3]{5})$, the rational prime $5$ ramifies as $(5)=\mathfrak{p}^3$, where $\mathfrak{p}=(5,\sqrt[3]{5})$, so you could consider the $\mathfrak{p}$-adic valuation on $K$.