Definition 1. Let be a field. An absolute value on is a map such that for all
- and iff
We will call the pair an absolute field. The absolute value that sends all nonzero elements to is called the trivial absolute value. One can also verify that the function defines a metric on Two absolute values are dependent if their induced topologies are the same, and independent otherwise. Note that
Hence and thus since Similarly
Proposition 2. Let and be two nontrivial absolute values on Then and are dependent iff If they are dependent, then there exist some such that
Theorem 3. (Approximation Theorem) (Artin-Whaples). Let be a field and be nontrivial pairwise independent absolute values on Then there exist elements such that for any there exists an such that
An absolute field is complete if all of its Cauchy sequences converge to a point in the field. One can show that each absolute field has a unique completion field up to isometry.
Definition 4. An abelian group is an ordered group if it has a partial ordering such that for all
Proposition 5. is an ordered group iff for a multiplicative subset where for all and
For an ordered group we will hereafter use to denote where and for all
Definition 6. Let be a field and be an ordered group. A valuation on is a map such that
We will call the triple a valuation field. The image of on is an ordered subgroup of Two valuations are equivalent if there is an isomorphism that respects order and value. One can also verify that and that if then
Definition 7. A subring of a field is a valuation ring if for all or
Proposition 8. If is a valuation ring, then it induces a valuation field.
Proof. Note that is a local ring, for, if are not units in then if we have
So if is a unit in , then which is a contradiction. Hence is not a unit. Furthermore if is not a unit, then for all we have that is not a unit. So the nonunits form an ideal in which is necessarily maximal (and uniquely so). So is a local ring. Let us denote this ideal by Thus we can write
where is the group of units of We now give a valuation. We will define our group as the quotient group In turn, we define with When then define iff and iff Observe that
Hence is a valuation field.
We thus hereafter refer to a valuation ring as the restriction of the induced valuation field to the ring.
Proposition 9. Let be fields where is a valuation field. Then there exists an extension making a valuation field.
Proof Idea. Let and We will call the valuation ring of One first takes the morphism and extends it to a valuation ring in One can construct an order preserving monomorphism
where is the maximal ideal in We then define as before, which is seen to agree with
Proposition 10. Let be fields and Let be a -valued valuation field and be the extension group to the induced valuation on Then
We will call the ramification index of in
Definition 11. A valuation is discrete if its codomain is a cyclic group.
It turns out that if is a valuation field and is the maximal ideal of its valuation ring, then there exists an element such that is a generator of Such an element is called a local parameter of One can also show that is a principal ideal generated by Moreover every element can be written as
for some unit integer is called the order of at If we say has a zero of order and if then we say has a pole of order
Proposition 12. Let be fields where is a finite extension of Further suppose that is a complete discrete valuation field (i.e. induced metric space is complete) and that and are the corresponding valuation rings and maximal ideals after extending the valuation. Then
 Lang, Serge. Algebra. Revised Third Edition. Springer-Verlag. 2000.