Let be a field, let be a discrete valuation on , and let be the valuation ring of . For each integer , define .
- Prove that is a principal ideal and that for all .
- Prove that if is a nonzero ideal of , then for some . Deduce that is a local ring with maximal ideal .
- It is clear that for all .
Now we show that is an ideal. We have , so is nonempty. Now let ; if one or both of and is 0, then . If , then . If , we have , so that . Moreover, , so that . Finally, if , we have . Then , so that . Since is commutative, we have that is an ideal of .
We now show that is principal. Choose such that ; such an element exists because is surjective. Note that exists in , and that . Let ; then . Note that , so that . Moreover, . Thus , and we have , and thus . In particular, is generated by any element of valuation .
- Let be a nonzero ideal of . Let be minimal among for , and let such that . In particular, we have . Moreover, since , we have . Thus .
From part (1), every proper ideal of is contained in ; thus is maximal, and moreover, is the unique maximal ideal of .