Exercise 5, § 8

Let K be a field, A a valuation ring of K and L a finite algebraic extension of K. Let A ⊃ A be another valuation ring of K; let A'_i (1 ≤ i ≤ m) be the valuation rings of L such that A'_i ∩ K = A' and let k'_i be their respective residue fields. Let k be the residue field of A' and A'' the valuation ring of K the canonical image of A; finally let A''_{ij} (1 ≤ j ≤ n_i) be the valuation rings of k'_i such that A''_{ij} ∩ k = A'' and A_{ij} the inverse images in A'_i of the A''_{ij} (1 ≤ i ≤ m, 1 ≤ j ≤ n_i).

(a) Show that the A_{ij} are valuation rings of L which are distinct and such that A_{ij} ∩ K = A and every valuation ring B of L such that B ∩ K = A is equal to one of the A_{ij}.

(b) Show that

and