Valuation Rings

Abstract

In §3 we have defined a valuation ring of a field K to be a subring A of K such that x ∈ A or x^-1 ∈ A for any non-zero x ∈ K. Obviously K is the quotient field of A, and K itself is a valuation ring of K. If K is absolutely algebraic (i.e., algebraic over its prime field) and of prime characteristic, then K is the only valuation ring of K (since any subring of K is a field). We shall see later that all other fields K have valuation rings distinct from K.