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Let $p$ be a prime number. A global field is defined as a finite extension of $\mathbb Q$ or $\mathbb F_p(t)$. On the other hand one can show that a local field (which is by definition a complete discrete valuation field with finite residue field) is a finite extension of $\mathbb Q_p$ or $\mathbb F_p((t))$.

Why do we use the adjectives "local" and "global"? What is the geometric picture that I am missing?

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    $\begingroup$ Global fields look like rings of meromorphic functions on "curves." Local fields look like rings of germs of meromorphic functions in a neighborhood of a point on a "curve." $\endgroup$ – Qiaochu Yuan Jan 19 '16 at 20:07
  • $\begingroup$ I think local is locally compact. Edit - en.wikipedia.org/wiki/Local_field seems to agree? $\endgroup$ – peter a g Jan 19 '16 at 20:08
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    $\begingroup$ @peterag, it is certainly true that "local fields" in the number-theoretic sense are locally compact, I don't think that is the origin of the modifier "local", but as QiaochuYuan notes, it is from physical local-ness. As in "local ring", which no longer entails local compactness. $\endgroup$ – paul garrett Jan 19 '16 at 20:15
  • $\begingroup$ @paulgarrett - Well, it is of course an etymological question... Do you know who coined the term (as opposed to the definition), or when? BTW, according to Wikipedia the term "compact" is due to Frechet in 1906. $\endgroup$ – peter a g Jan 19 '16 at 20:40
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    $\begingroup$ @peterag, I don't think Hensel in 1896 operated in the sort of abstraction that would have led him to talk that way, at all, although I may be wrong. (I don't think he used the specific word "local" or "lokalsche" or anything.) Also, Frechet's work seems not to have had much impact for a while after... I'd wager that the terminology is post-EmmyNoether, for example... $\endgroup$ – paul garrett Jan 19 '16 at 22:18
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In the early 1930s when local class field theory was developed, it was always called "class field theory in the small" (in German). Eventually the term local prevailed. In 1935, Hasse wrote, in a review of an article of Schilling, about the "local arithmetic structure of algebras". Later on, especially when articles in French and English on class field theory began appearing, the expression "local" was used, and the dichotomy small-large became local-global. In particular Chevalley's articles on local class field theory from 1938 use "theorie locale des corps de classes".

I don't think there's a big picture you're missing: in local fields, you always study one prime ideal, in global fieldsyou look at all of them at the same time. As a matter of fact, global fields can be characterized by a product formula (Artin-Whaples etc.).

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  • $\begingroup$ My vague memory is that e.g. "locally connected" was originally called "connected in the small" (in German, and maybe in some literal English translations from around the same period, maybe between the wars?). If so, then it would seem like "in the small" had the same meaning that "local" does now (in a n.h. of a point, more or less), and the transition from "in the small" to "local" was just a change of wording that doesn't reflect any real change in meaning. Do you think that's right? $\endgroup$ – tracing Jan 21 '16 at 14:58
  • $\begingroup$ "Class field theory in the small" = "Klassenkoerpertheorie im Kleinen"? Is that right? $\endgroup$ – peter a g Jan 22 '16 at 3:42
  • $\begingroup$ @tracing and peter: I would answer both questions by yes. $\endgroup$ – franz lemmermeyer Jan 22 '16 at 17:01
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Hopefully someone will correct me if I am wrong. In algebraic geometry over an algebraically closed field $k$, we have a notion of looking at things locally. This is done by looking at a stalk. If $X$ is an affine variety (by definition, an irreducible closed set in $n$-space $k^n$), and $R$ is the coordinate ring of $X$ (by definition, $k[T_1, ... , T_n]$ modulo the prime ideal corresponding to $X$), then we have a notion of a ring $\mathcal O_X(U)$ of regular functions for each open set $U$ of $X$. By definition, the elements of $\mathcal O_X(U)$ are functions from $U$ to $k$ which locally can be written as rational functions in $k$.

Now, if $x$ is a point of $X$, the stalk $\mathcal O_{X,x}$ is the ring of equivalence classes $(f,U)$, where $U$ is a neighborhood of $x$ and $f \in \mathcal O_X(U)$, where two classes $(f,U)$ and $(g,V)$ are the same if $f$ and $g$ agree on some neighborhood of $x$ which is contained in $U \cap V$. The upshot is that $\mathcal O_X(X)$ is canonically isomorphic to $R$, and if $\mathfrak m$ is the maximal ideal of $R$ corresponding to the point $x$, then $\mathcal O_{X,x}$ is canonically isomorphic to $R_{\mathfrak m}$. This is where 'localization' gets its name, because from the geometric side we are literally looking at the local property of functions.

Instead of looking at a variety $X$, you can create similar parallels by looking at an arbitrary commutative ring $B$ and its associated 'affine scheme,' but I don't think these notions were popular until Grothendieck.

Now if $K$ is a number field with ring of integers $A$, then many results about the structure of $A$ are determined by looking at its prime ideals, which can be isolated by localizing. Now if $P$ is a prime ideal of $A$, and $v$ is the absolute value corresponding to $P$, then many properties about $A$ are connected to the completed ring $A_v$.

Since $P$ completes to the unique maximal ideal of $A_v$, it is natural to refer to the quotient field of $A_v$ as a local field, because its properties carry over to properties about the localized ring $A_P$.

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In addition to the other answers one could give a number-theoretic "picture", e.g., the Hasse–Minkowski theorem: A quadratic form over a number field nontrivially represents zero if and only if this holds for all completions of the field. So we have a nontrivial solution, say, over the global field $\mathbb{Q}$ if and only we have a nontrivial solution over all local fields $\mathbb{Q}_p$, $p$ prime and $p=\infty$. This illustrates the meaning of "local" and "global" with respect to solutions of polynomial equations ( see also the "local-global" principle in general, also discussed here). This may be more concrete than a "geometric picture".

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