Vol. 45, No. 1, 1973

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Infinite primes of fields and completions

David Kent Harrison and Hoyt D. Warner

Vol. 45 (1973), No. 1, 201–216

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.

Mathematical Subject Classification 2000
Primary: 12J10
Received: 27 October 1971
Published: 1 March 1973
David Kent Harrison
Hoyt D. Warner