This is not quite the case in my experience. What I've seen is that a number field is defined as any finite extension field of Q in C. Rather, when people talk about factorization, primes, etc. with respect to a number field, they are talking about those things in the number field's ring of algebraic integers (those elements of that number field that are roots of some integer polynomial that's monic. --Dan
On Dec 3, 2014, at 2:58 PM, Gareth McCaughan <gareth.mccaughan@pobox.com> 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.