Extending the p-adic norm

Question

I am reading the book p-adic Numbers, p-adic Analysis, and Zeta-Functions by Koblitz, and I am trying to understand the proof of the theorem on page 61:

Let $K$ be a finite field extension of $\mathbb{Q}_p$. Then there exists a field norm on $K$ which extends the the norm $\vert \hspace{0.5em} \vert_p$ on $\mathbb{Q}_p$.

One of the things that is first mentioned in the proof is that the function \begin{array}{rcl}\vert\alpha\vert_p:=\vert\mathbb{N}_{K/\mathbb{Q}_p}(\alpha)\vert^{\frac{1}{n}}_p\end{array}is multiplicative, where the norm on the RHS is the usual $\vert \hspace{0.5em} \vert_p$ on $\mathbb{Q}_p$ and $n=[K:\mathbb{Q}_p]$. This is the part that confuses me.

By definition, $\mathbb{N}_{K/\mathbb{Q}_p}(\alpha\cdot \beta)= (\mathbb{N}_{\mathbb{Q}_p(\alpha\cdot \beta)/\mathbb{Q}_p}(\alpha\cdot \beta))^{[K:\mathbb{Q}_p(\alpha\cdot\beta)]}$. For the function to be multiplicative, I would need the equality $$\mathbb{N}_{\mathbb{Q}_p(\alpha\cdot \beta)/\mathbb{Q}_p}(\alpha)=\mathbb{N}_{\mathbb{Q}_p(\alpha)/\mathbb{Q}_p}(\alpha).$$However, it is not in general true that $\alpha\in\mathbb{Q}_p(\alpha\cdot\beta)$ so the above equality does not make sense. What am I missing?

Answer 1

Use another definition of the norm $N_{K/\mathbb Q_p}(x)$ for $x \in K$. It is by definition the product of all $\sigma(x)$, where $\sigma$ runs through all $\mathbb Q_p$-embeddings of $K$ into $\overline{\mathbb Q_p}$. It is obvious by this definition that $N_{K/\mathbb Q_p}(xy) = N_{K/\mathbb Q_p}(x) N_{K/\mathbb Q_p}(y)$ for all $x, y \in K$.

Answer 2

The definition of the norm that you use on a finite extension of fields is terrible. See Definition 2.8 here for a much better definition, and Theorem 3.2 there shows easily with the better definition that the norm map is multiplicative. The equivalence of the better definition and the lousy definition you are using is in Equation (5.2). Corollary 5.15 there relates the better definition of the field norm with the description in terms of a product over conjugates (with multiplicity).