**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

.

**Prove:**
(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