Valuation Ring - Value Group

Value Group

The units D* of D comprise a group under multiplication, which is a subgroup of the units F* of F, the nonzero elements of F. These are both abelian groups, and we can define the quotient group V = F*/D*, which is the value group of D. Hence, we have a group homomorphism ν from F* to the value group V. It is customary to write the group operation in V as +.

We can turn V into a totally ordered group by declaring the residue classes of elements of D as "positive". More precisely, V is totally ordered by defining if and only if where and are equivalence classes in V.