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I just came across the p-adic norm for the first time. I tried to show that it is actually a norm on $Q$ but I was asking myself, whether checking scalability is a bit self referential ?

What I mean by that is, that I am used to show scalability under the absolute value for any scalar from the underlying field (mostly due to my lack of experience this was in the context where there is a clear intuitive interpretation for the absolute value, e.g. on $\mathbb{R} \quad or \quad \mathbb{C}$, but with the p-adic the scalability criterion is only satisfied w.r.t the p-adic itself which seems self referential to me. Can somebody help clarify this notion ?

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I'm not sure what you mean. The standard definition of a norm on $\mathbb{Q}$ (or any field) is a function $| \cdot |: \mathbb{Q} \rightarrow \mathbb{R}^{\geq 0}$ such that $|x| = 0 \iff x= 0$, $|xy| = |x| |y|$ and $|x+y| \leq |x| + |y|$. I don't see the "scalability" of which you speak... – Pete L. Clark Jan 30 '12 at 20:04
I am used to checking if $(F,|.|)$ is the underlying field that for all $ a \in F $ and $ x \in X $ that $|| a x || = |a| \cdot ||x||$ but for the p-adic I end up showing $ ||a x|| = ||a|| ||x|| $ and it seems to me whilst I m trying to show ||.|| is a norm I can t use it as one on my underlying field $F$ yet ?? – Beltrame Jan 30 '12 at 20:09
I think you are confused between the concept of a norm $|| \cdot ||$ on a vector space $V$ over a normed field $(K,|\cdot|)$ and a norm $|\cdot |$ on a field $K$. In the latter case, one of the conditions is indeed $||av || = |a| ||v||$ for all $a \in K$ and $v \in V$, but that is not your situation when you are examining the $p$-adic norm on $\mathbb{Q}$. – Pete L. Clark Jan 30 '12 at 20:18
up vote 2 down vote accepted

Sure, what you call the "scalability criterion" is indeed "self-referential," but it's not a problem because the axioms which define field norms are perfectly well-defined and are not intended to determine the norm uniquely. A norm here is any function $|\cdot|:K\to \mathbb{R}^{\ge0}$ such that $|x|=0$ iff $x$ is zero, satisfies what is likely better termed as multiplicativity i.e. $|xy|=|x||y|$, and the triangle inequality $|x+y|\le|x|+|y|$. The issue you seem to have is that you're confusing a vector space norm, which presumes an absolute value on the underlying base field is already understood, and an out-in-the-wild field norm, where no such presumptions are taken for granted and we're defining something directly on the field itself instead of some overarching structure built on top of the field.

So to check the multiplicativity of the $p$-adic norm, you need only invoke unique factorization...

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great, thanks ! The absolute value norm was indeed mostly something that was always taken for granted, so your answer makes it very clear how to get started with checking some criteria for a norm on the Field. If I understand you correctly, this will also get rid of any self referential issues right ? – Beltrame Jan 30 '12 at 20:24
@PeeJay: What "issues" are there to get rid of? It's a function that satisfies so-and-so properties. When, for example, a linear map is defined such that $f(a+b)=f(a)+f(b)$ and you're given the exercise to check that a particular $f$ is linear, surely you don't get bogged down saying that linearity is self-referential? – anon Jan 30 '12 at 20:27
absolutely, I agree. But this was not apparent to me before I understood, that to get off the ground with a norm on the field you don t really need an underlying one to check scalability or multiplicativity as you put it. – Beltrame Jan 30 '12 at 20:35

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