**Result: ** is a discrete valuation satisfying the following properties:

- is surjective.

Then for some prime , given by, for .

**Proof: **It is a fact (cf. Dummit & Foote Ex 39 Sec. 7.4) that is a local ring with a unique maximal ideal of elements of positive valuation. (Recollect that an element of is a unit iff its valuation is zero.) Now, .

Claim: . Clearly, being surjective, because otherwise, each nonzero integer would have valuation zero and so would every nonzero rational. If then we factorize with integers. Then or . Thus must be a prime ideal. So the claim is justified.

Now given , write with . I claim that and are units. Since , we can write . If is not a unit then and thus , a contradiction. A similar argument suggests that is also a unit. Hence,

Now surjective implies, there exists such that . This leaves two possibilities, namely, . But gives an easy contradiction: . Thus the only possibility is that and thus, .

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