The notion of infinite prime in a
ring with identity, defined in the first author’s memoir “Finite and infinite primes for
rings and fields” (A.M.S Memoir # 68), is studied in fields. Extending results of R.
Baer and D. W. Dubois, each infinite prime P of a field F is shown to determine a
complex place ϕP of F such that ϕP(P) is the set of nonnegative reals in ϕP(F),
and is an infinite prime of ϕP(F). The collection of all infinite primes P
of F determining the same ϕ and ϕ(P), is shown to be describable, in an
almost purely multiplicative way, in terms of certain groups determined by p
and ϕ(P). Using these theorems a notion of completion of a field at a finite
or infinite prime is given, generalizing the classical notion for the prime
divisors of a number field. These completions are characterized as certain
linearly compact fields and are shown to be in general unique only when P is
real (i.e., ϕP(F) is contained in the reals). For a fixed prime P of F, the
set of elements of F which are squares in every completion of F at P is
calculated.
A characterization of number rings is given and examples of pathology in infinite
primes are indicated.
