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Ok so I've seen the adjective 'arithmetic' (stress on the e) bandied about from time to time in reference to the "arithmetic theory of some subject" (elliptic curves for instance), or the "arithmetic side of things", what do people mean when they use this adjective? Surprising as it may seem, I have not been able to divine its meaning from context. Thanks.

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It means number-theoretic, as opposed to working over an algebraically closed field, or a real-closed field.

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Ok thanks André, so by number-theoretic you mean dealing with the problem of finding the subset of a polynomial's roots which happen to lie in your non-algebraically closed field? Also, are there algebraically closed fields which do not contain the algebraic numbers? – Thoth Oct 21 '12 at 17:38
Typically, one might be interested in the integer solutions, or the rational solutions, or the solutions over a finite extension of the rationals. Or else one is working over a polynomial ring, with restricted coefficients. Much more complicated structures may be involved, but always with a number-theoretic component. "Arithmetic," among other meanings, is an old-fashioned term for Number Theory. – André Nicolas Oct 21 '12 at 17:46

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