Let be a field. A discrete valuation on is a function satisfying the following.
- is surjective
- Provided , .
The set is called the valuation ring of .
- Prove that is a subring of which contains the identity.
- Prove that for each nonzero element , either or .
- Prove that is a unit if and only if .
First, we note a few facts about .
If , then . Since is surjective, is an arbitrary integer. By the uniqueness of additive identities, . Now , so that . Thus . Finally, if , , so that .
Now we show that is a subring. by definition, so that is not empty. Let and consider . If and , then . If and , then . If , then . If , then either or , and then . Now consider . If , then . If , then , so that . Thus is a subring of . As we saw earlier, , so that .
Suppose is nonzero. Now . Thus , and either or is nonnegative.
Suppose is a unit; then . Above, we saw that , and both and are nonnegative. Thus . Now suppose . Then , so that and is a unit.