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Number theory is also sometimes just called "arithmetic," i.e., as in Serre's A Course in Arithmetic.

This isn't a very widespread use of the word, though, and I don't think even Serre intended the two to be synonymous. I always took it more in the sense that arithmetic was the study of elementary (as in fundamental, not as in easy) properties of rational numbers, be it the standard rational numbers or the p-adic ones. Usually if one uses the word "arithmetic" with respect to modern number theory, it's used as an adjective, as in "arithmetic algebraic geometry."


Surely "arithmetic" is the study of the algebraic properties which arise as a result of a ring or field not being algebraically closed. Hence the common use of phrases such as: "We work over an algebraically closed field so that issues of arithmetic do not arise".

Conversely "number theory" is not confined to algebraic properties but is normally rstricted to (ground) rings and fields of transcendence degree 0.

Back in the days of Euclid and Diophantus, "arithmetic"
referred to the study of properties of natural numbers
and formed part of the quadrivium, the mathematical part
of the curriculum throughout the Middle Ages. It was only
much later when the quadrivium faded away and textbooks
on the rudiments of the subject for little kids came out
that the semantic shift happened. To get some idea how
recent this is, remember that Gauss used the term "Arithmetic"
and, as pointed out, this usage is still found with Serre
and Hardy (who had to relent in the title of his book
because of fear of popular misunderstanding but who uses the
term inside of it). In order to avoid confusion with simple
exercises in adding and multiplying numbers, the term
"higher arithmetic" is used for the study of more advanced
properties of the integers.

Arithmetic in the form that medievals and Euclid used was born 1,700 years earlier. Ancient Near East scribes generally converted rational numbers by three-steps, obtaining optimal or elegant unit fraction series. Knowledge of aliquot parts of denominators, the second-step, was written in additive numerators, was well-known to Euclid.

Additional properties of Euclid' arithmetic were erased by the arrival of decimals, 400 years ago. Today, rational numbers are rounded-off in ways that may be making Archimedes, Euclid and Diophantus to roll-over in their graves.

What exactly to you mean "normally restricted to (ground) rings and fields of transcendence degree 0?" If you're considering a field as a "ground" field, to speak of its transcendence degree makes no sense as it implies you have another "ground" field in mind; and it is certainly not true that number theory is restricted to algebraic extensions of ground fields (for instance the fields \mathbb{C}(t) and \mathbb{F}_p(t) are of interest in number theory).

Even more to the point, arithemeticians (a.k.a. number theorists)
are certainly interested in transcendental numbers. For instance,
consider the formula expressing pi as product involving prime
numbers, formulas for prime numbers by truncating powers of
real numbers, questions about the transcendance of logarithm
and exponents as well as various constants which appear in
analysis, the recent interest in periods, continued fractions of
transcendental numbers, etc. A good part of analytic number
theory and Diophantine approximation deals with such issues
regarding transcendental numbers, so I would say that the real
number field is not just something useful in the study of
algebraic numbers but a primary object of study in higher arithmetic.

Sorry, by transcendence degree 0 I meant over the subring additively generated by 1.

None the less your first example shows that I was talking complete rubbish as if you take \mathbb{C} as your ground ring for
\mathbb{C}(t) then \mathbb{C} contains transcendental numbers such as pi.

I still think my interpretation of "arithmetic" is the conventional one though.

I figured that was the reason the term "higher arithmetic" arose in the first place, to avoid confusion. I can cite three different books titled "Higher Arithmetic" that are about number theory. For that reason I thought at least a remark was necessary.

P.S. Thanks to everyone else who weighed in on this.

Oh I totally agree it's worth a remark. I wasn't suggesting anything was wrong with your entry, I was just making a comment. Go number theory.

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