Background: Let be a field. A discrete valuation on is a function on is a function such that
(i) for all
(ii) is surjective
(iii) For all such that , min . The set is called the valuation ring of .
(a) Prove that is a subring of .
(b) Prove that for each nonzero element , , or .
(c) Prove that is a unit if and only if .
(d) Now let and let be a prime. Given a nonzero element , write where is an integer and divides neither nor ; then define . Prove is uniquely defined and that is a discrete valuation on