Discrete valuation
Encyclopedia
In mathematics
, a discrete valuation is an integer
valuation on a field
k, that is a function
satisfying the conditions
.
Note that often the trivial valuation which takes on only the values
is explicitly excluded.
A field with a non-trivial discrete valuation is called a discrete valuation field.
we can associate the subring
of
, which is a discrete valuation ring
. Conversely, the valuation
on a discrete valuation ring 
can be extended to a valuation on the quotient field 
giving a discrete valued field 
, whose associated discrete valuation ring 
is just 
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a discrete valuation is an integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
valuation on a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
k, that is a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
satisfying the conditions
.Note that often the trivial valuation which takes on only the values
is explicitly excluded.A field with a non-trivial discrete valuation is called a discrete valuation field.
Discrete Valuation Rings and valuations on fields
To every field with discrete valuation we can associate the subring
of
, which is a discrete valuation ringDiscrete valuation ring
In abstract algebra, a discrete valuation ring is a principal ideal domain with exactly one non-zero maximal ideal.This means a DVR is an integral domain R which satisfies any one of the following equivalent conditions:...
. Conversely, the valuation
on a discrete valuation ring can be extended to a valuation on the quotient field giving a discrete valued field , whose associated discrete valuation ring is just .Examples
- For a fixed prime
for any element 
different from zero write 
with 
such that 
does not divide 
, then 
is a valuation, called the p-adic valuation.
See also
- discrete valuation ringDiscrete valuation ringIn abstract algebra, a discrete valuation ring is a principal ideal domain with exactly one non-zero maximal ideal.This means a DVR is an integral domain R which satisfies any one of the following equivalent conditions:...
- Valuation (mathematics)Valuation (mathematics)In algebra , a valuation is a function on a field that provides a measure of size or multiplicity of elements of the field...
- Valuation ringValuation ringIn abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F, at least one of x or x −1 belongs to D....