TITLE: Norms on fields
QUESTION [6 upvotes]: I'm doing an introductory module in number theory, and came across the definition of a norm on a field. It seems to agree with the definition of a norm on a vector space over a field (just view the field as a vector space in the former definition). Then I read Ostrowski's Theorem that says 'every non trivial norm on $\mathbb{Q}$ is equivalent either to the standard absoltute value or to the p-adic norm for some prime p. All these norms are inequivalent.'
I don't see how this theorem doesn't contradict the theorem that all norms on a finite dimensional vector space are equivalent.....However I see a difference in the way 'equivalence' is being defined between norms in the case where the norms are on a field as opposed to a general vector space. Everything seems a bit all over the place in my head and I don't know why there is a difference in definition of norms being equivalent on fields and norms on vector spaces when surely a field is a vector space...
If someone could clear things up I'd appreciate it.

REPLY [4 votes]: I think that everything that @Eero has said is true, but there’s more wrong in the situation than that.
You have not quoted the theorem about norms on a finite-dimensional vector space fully. First, you have to start with a normed field $k$, and ask about the norms on a finite $k$-space that in some sense extend that norm. Second, the theorem does not hold if $k$ is not complete under the given norm.
So there’s no contradiction, because in talking about $\Bbb Q$, you have not started with a pre-given norm on a field. Rather, you are asking whether there may be more than one norm on a field that has been given abstractly. Since $\Bbb Q$ has many topologies on it, each of them making it a topological field (with continuity of $+$, $\times$, and $\div$, for instance) and all of them different as topologies, it should not be surprising that the completions are different.