**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

and

Hence and thus since Similarly

and

**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

for all

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

- iff

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

and

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

[1] Lang, Serge. *Algebra*. Revised Third Edition. Springer-Verlag. 2000.