Rich Schroeppel wrote:
Note for the uninitiated: When number theorists use the phrase "Number Field", they mean a ring, not a field. The number field generated by sqrt2 includes the algebraic integers A+Bsqrt2 with A & B ordinary integers, but does not include 1/2, or 1/3, or 1/sqrt2. This terminological nuance (to use a polite phrase) helps us separate out our special clique.
I wonder whether this nuance is characteristic of some sub-clique rather than of number theorists as a whole. Both the algebraic number theory textbooks I just checked (Lang and Frohlich&Taylor) define a "number field" to mean a finite extension of the rationals. So (for what little it's worth) do Wikipedia and MathWorld. Everything on the first page of hits for "number field" on MathOverflow uses the term to refer to the field rather than its ring of integers. So, in fact, does everything else I can easily lay my hands on. (Some of them prefer the term "algebraic number field", and some of those explicitly say something like "algebraic number field, or number field for short".) I don't have much in the way of actual scholarly articles written by actual number theorists to hand, but I tried the experiment of looking at recent articles in the Journal of Number Theory; all I can see is their abstracts, but they too seem to be unanimous in using the term to mean the field rather than its ring of integers. -- g