I apologize if I'm posting the same comment that I already posted a couple of months ago. (I can't recall if I actually posted it or just intended to do so.) But: I don't think number theorists mean a non-field ring when they say "number field". Rather, they mean an actual field K that is a finite extension of the rationals Q (an "algebraic" extension of Q). But then attention is often directed to the ring O(K) of algebraic integers of K. O(K) is defined as all members of K that are roots of monic polynomials with coefficients in Z. It is not immediately obvious that for all algebraic extensions K of Q, the set O(K) is always closed under addition and multiplication -- i.e., is actually a ring -- but it is. --Dan
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.