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I think (discrete) valuations come up very naturally in the theory of meromorphic functions/Laurent series, and by analogy the theory of $p$-adic numbers, and generalizations of those. One can then develop the nice theory of discrete valuation rings (e.g. Euclidean domain, a unique maximal ideal, uniformizing elements, whatnot), thereby elevating discrete valuations to be functions of importance.

Then, exponentiating one such valuation yields an absolute value, which can then produce a metric, and hence a bunch of analysis/topology. My issue is that the multiplicativity of an absolute value is not necessary in creating a metric, and so I personally feel this approach conflates two separate ideas into one.

So my question is this: why is it natural/important to exponentiate the valuation to produce a multiplicative function? Choosing (normalizing) a "canonical" base on which to exponentiate is explained here using the ideas of Haar measure (also in a comment by KCd in Why does the p-adic norm use base p?), so I'm really asking about the naturality of the exponentiating the discrete valuation itself, and not the naturality of the base.

I also know that absolute values (or more precisely their product formula/identity) can characterize global fields (Artin-Whaples). So,

I suppose absolute values are very important somehow; but can somehow give an intuition/heuristic that explains why multiplicativity/absolute values/product formulas play such an important role here?

I guess my main issue is that from an analyst's perspective, the main tool for the standard absolute value on $\mathbb R$ is the triangle inequality and positivity, not multiplicativity, so perhaps I just don't have an appreciation for what multiplicativity can do.

EDIT: one idea I had is that in this Quora post, it is mentioned that "in the best possible cases (such as what happens in the Hasse-Minkowski theorem), the problem is solvable for rationals if and only if it is solvable for the reals and all of the p-adics." (I think the $\mathbb Q$-version of Hasse-Minskowski does not need mention of absolute values), so this may provide the motivation to look into why $\mathbb R$ and $\mathbb Q_p$ are "good" completions of $\mathbb Q$, and the development of the adele ring. The Quora post also mentions the Fourier theory, which is nicely developed for locally compact abelian groups (I guess because of Haar measure, which, according to the Wiki for local fields, can be used to construct an absolute value generating the topology). This reminded me of the definition of a local field, i.e. that being a locally compact topological field is essentially equivalent to being a completion of some field $K$ w.r.t. some absolute value. So perhaps this locally-compactness is why although there are many Completion of $\mathbb{Q}$ other than $\mathbb{R}$ and $\mathbb{Q}_p$?, the adele ring Wikipedia page and above Quora post both say "all the completions", to mean all the completions that are "good" (in the sense that they admit Fourier theory $\approx$ they are completions w.r.t. an absolute value?)

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  • $\begingroup$ You write "the main tool for the standard absolute value on $\mathbf R$ is the triangle inequality and positivity, not multiplicativity" but think about how you estimate terms in power series: $|a_nx^n| = |a_n||x|^n$ is quite useful! The very idea of power series involves multiplication in $\mathbf R$. There aren't power series in abstract metric spaces. $\endgroup$
    – KCd
    Commented Mar 29, 2022 at 20:07
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    $\begingroup$ "I've never really felt the need to develop power series on spaces other than ℝ or ℂ": then I guess you've never worked with complete valued fields other than the real and complex numbers. Analytic functions of a $p$-adic variable are quite useful in number theory, and they provide many reasons to make use of multiplicativity of the absolute value. In classical analysis you also have Banach algebras, where the absolute value is submultiplicative. If you care about spaces on which there is a multiplication, then you want it to interact well with the absolute value/norm/etc. $\endgroup$
    – KCd
    Commented Mar 30, 2022 at 4:15
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    $\begingroup$ If you want to make a tensor product of Hilbert spaces into a Hilbert space in a useful way then you want the inner product of elementary tensors to be related in a nice way to inner products on the original Hilbert spaces. Inner products on a real or complex vector space are a kind of multiplication (a bilinear operation) with some nice additional properties. $\endgroup$
    – KCd
    Commented Mar 30, 2022 at 4:22
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    $\begingroup$ They're used in the proof of Skolem's theorem, which can provide a method of bounding the number of integral solutions to some Diophantine equations. An example: kconrad.math.uconn.edu/blurbs/gradnumthy/x3-2y3=1.pdf. I still find the idea in your post that multiplication is not an essential concept in classical analysis to be rather strange. Essentially all series expansions (Fourier, Dirichlet) and integral transforms (Fourier, Laplace) involve multiplication of terms, and the simplest way to estimate such things requires knowing how the absolute value behaves on a product. $\endgroup$
    – KCd
    Commented Mar 31, 2022 at 20:45
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    $\begingroup$ Even if classical analysis can be developed on spaces that don't have an algebraic structure, we probe the spaces by working with nice real or complex-valued functions on them, and you're going to be multiplying those function values by numbers a lot. $\endgroup$
    – KCd
    Commented Mar 31, 2022 at 20:46

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