Valuation (algebra)

In the mathematical subfield of valuation theory it comes to generalizations of the question of which power of a fixed prime number is a natural number divisible.

• 2.1 Definition
• 2.2 Examples
• 2.3 Discrete Reviews and discrete valuation rings

• 3.1 Definition
• 3.2 Reviews and Rating rings

P- Review

It is a prime number.

The evaluation ( also: the -adic valuation, or exponent ) of a natural or whole number is the largest number such that there is still divisible by. The rating indicates how often a prime number in the prime factorization of a natural or integer.

The evaluation of a natural number is the exponent of the prime number in the prime factorization of. is

So is

If a prime number not in the prime factorization of, then.

It is because any power of each prime number which divides 0.

The evaluation of an integer is the sum of her.

The evaluation of the rational number, the difference of the evaluations of the numerator and denominator: a rational number having being so

If p only in the denominator of (fully truncated ) break, so is a negative number.

The evaluation of rational numbers play an important role in the type of construction of a p-adic number: function

Forms on the rational numbers a non- Archimedean amount.

All p- and S- integers

An - integer (also " adic integer " or " for all speed") is a rational number, the non-negative - assessment, ie when in a completely reduced fraction representation of the denominator is not divisible by. Rational numbers that are not-quite, are sometimes also called " broken".

The set of all integers, is a sub- ring that is written. is a discrete valuation ring, in particular, there are up to Associated exactly an irreducible element, namely.

Is generally a set of prime numbers, an integer - is a rational number, the whole- for each (? ), Ie when in a completely reduced fraction representation of the denominator is only divisible by primes. The set of - integers form a subring of. For example, for so is

Discrete reviews

Definition

It is a body. Then is called a surjective function

A discrete valuation, if the following properties are satisfied:

For everyone. together with means discrete valued body.

Examples

• The valuation on the rational numbers for a prime number
• The zeros of meromorphic functions or Polordnung in a fixed point

Discrete Reviews and discrete valuation rings

The subset

Forms a subring of the valuation ring of. He is a discrete valuation ring with maximal ideal, which is a principal ideal.

Conversely, if a discrete valuation ring, then by

A discrete valuation on the quotient field of defined.

Discrete valuation rings and discretely evaluated body correspond to each other.

General reviews

Definition

Is a totally ordered abelian group and a ( commutative ) field, then a mapping

An assessment if the following properties are satisfied:

For everyone.

Then is called a weighted body with value group.

Reviews and Rating rings

An integral domain is called valuation ring if it has the following property:

Is a valuation ring with quotient field, then one can define a valuation on a set of values ​​:

This refers to the image of in; order on is defined by

Conversely, if a weighted body with evaluation, is

An evaluation ring, which is then called the evaluation ring for evaluation. The group is canonically isomorphic to the group of values ​​.

For a body, it gives up and review rings a bijective relationship between isomorphism classes of reviews that are included.