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In scheme theory, an algebraic scheme is the data of a scheme + a morphism of finite type to the spectrum of a field. Where does the term "algebraic scheme" come from? It does not seem intuitive to me (to me all schemes are equally algebraic, others may have a different opinion). Are there any historical accounts regarding this matter?

Here is one mention of the category of algebraic schemes without an explicit reference to the base (though it appears that in that terminology all schemes are understood to come with a morphism to the spectrum of a field). In the Stacks Project, they use "algebraic $k$-schemes" (with $k$ being a field).

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    $\begingroup$ It's not a great terminology, and I actually never saw it (or didn't pay attention). +1 if you prefer "scheme of finite type over a field". $\endgroup$
    – YCor
    Commented Jun 7, 2019 at 7:54
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    $\begingroup$ But actually now that I'm looking, I can see in Grothendieck (1960) numdam.org/article/SB_1958-1960__5__193_0.pdf the use of "schéma algébrique sur $A$" (algebraic scheme over $A$), where $A$ does not have to be a field (yet it seems to mean finite type). $\endgroup$
    – YCor
    Commented Jun 7, 2019 at 7:57
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    $\begingroup$ In the context of field extensions, finite type does not imply algebraic. $\endgroup$
    – François Brunault
    Commented Jun 7, 2019 at 17:03
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    $\begingroup$ finite type schemes over a field are exactly the schemes which correspond to algebraic varieties, so algebraic scheme = algebraic variety. I think that is the reason for the name. $\endgroup$
    – display llvll
    Commented Jun 9, 2019 at 12:59
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    $\begingroup$ The reason is that an algebraic scheme over a field can be covered by a finite number of spectra of finitely generated algebras. Those affine schemes that arise as the spectrum of a finitely generated algebra over a field are called algebraic, and the intuition behind this is that they are defined by a finite set of polynomials in a finite number of symbols. I think this terminology and point of view is common in the context of group schemes, see for example the notes of J.S. Milne $\endgroup$
    – display llvll
    Commented Jun 9, 2019 at 16:36

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