Example (4)

Let A be a principal ideal domain, K its field of fractions and p an extremal element of A. For x ∈ K* let v_p(x) denote the exponent of p in the decomposition of x into extremal elements (Algebra, Chapter VII, § 1, no. 3, Theorem 2); it is immediately seen that v_p is a valuation whose order group is Z and whose ring is A_{A_p}. By Proposition 3 of § 1, no. 4 we thus obtain, up to equivalence, all the valuations on K which are not improper and are positive on A. Taking A = Z we recover the p-adic valuations on Q (General Topology, Chapter IX, § 3, no. 2); these valuations are, up to equivalence, the only valuations on Q which are not improper (§ 1, no. 4, Corollary 1 to Proposition 3). Taking A = k[X], where k is a field, the non-improper valuations on k(X) whose restrictions to k are improper are (up to equivalence): on the one hand the valuations v_P where P runs through the set of irreducible monic polynomials of k[X] and on the other hand the valuation v defined by

for P ∈ k[X] and Q ∈ k[X] (§ 1, no. 4, Corollary 2 to Proposition 3); all these valuations obviously have Z as order group and their residue fields are monogenous algebraic extensions of k (Algebra, Chapter V, § 3, no. 1).