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In study of algebraic number theory one often comes across the terms 'infinite' and 'finite' places, referring to the archimedean and non-archimedean valuations of your field, respectively - but I have no intuition as to why they're called that! What's the motivation for this terminology (for example, why is the standard absolute value said to come from the 'infinite prime')? Can it be tracked back to some author in particular?

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  • $\begingroup$ would this involve infinite decimal places vs finite decimal places (or at least countably infinitely many)? Just a guess $\endgroup$ – Don Larynx Nov 14 '13 at 2:25
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It is because of the analogue in the function-field case, that is where $\mathbb Z$ becomes $\mathbb F_q[x]$. The latter has the obvious primes, but/and in projective one-space, the "point at infinity" corresponds to the ideal generated by $1/x$ in (the valuation ring obtained by localizing) $\mathbb F_q[{1\over x}]$. One point is that the ideal corresponding to the point is not an ideal of the original ring, but of a different related ring. That valuation, attached to the point at infinity, on the fraction field $\mathbb F_q(x)$ is the only one not given by an ideal in $\mathbb F_q[x]$. Thus, by analogy, since the "usual" metric on $\mathbb Q$ is the only metric not given by an ideal in $\mathbb Z$ (all the latter are the $p$-adic ones), we might imagine that it corresponds to some mythical (point at) infinity.

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