Exercises. 1.1. Let D be a unique factorization domain, and let E D be an irreducible element. Show there is a unique valuation v: D+RU{c} satisfying v(f) = 1 and v(g) = 0 for any irreducible g not associate to f. 1.2. Let u be a valuation on a commutative ring R. Show that w: RT] RU{c} defined by wlao + ... + a,T") = min{v(ao),..., v(@r.)} is a valuation on R[T] extending v. 1.3. Let k be a field, and let v be the extension to k(T) of the T-1-adic valuation on k[T-4). Describe the restriction of v to k[T]. 1.4. Construct two nonequivalent valuations v,w on Z[T] satisfying v(T) = x(T) = 1. 1.5. Classify all equivalence classes of valuations on 2. 2. Valued fields and the valuation subring DEFINITION 2.1. A valued field is a field equipped with a valuation. We say a valuation v on a field K is discrete if v(K) is a discrete subgroup of R. 2. VALUED FIELDS AND THE VALUATION SUBRING 14 Exercises. 1.1. Let D be a unique factorization domain, and let E D be an irreducible element. Show there is a unique valuation v: D+RU{c} satisfying v(f) = 1 and v(g) = 0 for any irreducible g not associate to f. 1.2. Let u be a valuation on a commutative ring R. Show that w: RT] RU{c} defined by wlao + ... + a,T") = min{v(ao),..., v(@r.)} is a valuation on R[T] extending v. 1.3. Let k be a field, and let v be the extension to k(T) of the T-1-adic valuation on k[T-4). Describe the restriction of v to k[T]. 1.4. Construct two nonequivalent valuations v,w on Z[T] satisfying v(T) = x(T) = 1. 1.5. Classify all equivalence classes of valuations on 2. 2. Valued fields and the valuation subring DEFINITION 2.1. A valued field is a field equipped with a valuation. We say a valuation v on a field K is discrete if v(K) is a discrete subgroup of R. 2. VALUED FIELDS AND THE VALUATION SUBRING 14