Assessment (algebra)

Values ​​of bodies are important in body theory , a field of algebra . Non-Archimedean p-adic evaluations are used to construct the p-adic numbers and are therefore fundamental to p-adic geometry. In older approaches to algebraic geometry , evaluations of function fields were also used.

reviews

An evaluation of a body is a function in an arranged body with the properties ${\displaystyle K}$ ${\displaystyle \varphi \colon K\to P}$ ${\displaystyle P}$

1. ${\displaystyle \varphi (x)\geq 0}$ and ${\displaystyle \varphi (x)=0\iff x=0}$
2. ${\displaystyle \varphi (xy)=\varphi (x)\varphi (y)}$
3. ${\displaystyle \varphi (x+y)\leq \varphi (x)+\varphi (y)}$

An example of a valuation is the amount function on the real or complex numbers with the signature . A rating is called non-Archimedean if for . It can be shown that an evaluation is non-Archimedean if and only if it satisfies the tightened triangle inequality . In number theory today, however, the non-Archimedean exponential evaluations defined below are usually meant when "evaluations" are mentioned. ${\displaystyle |x|}$ ${\displaystyle |\cdot |\colon \mathbb {C} \to \mathbb {R} }$ ${\displaystyle |\cdot |\colon K\to \mathbb {R} }$ ${\displaystyle |n|\leq 1}$ ${\displaystyle n\in \mathbb {N} }$

See also : Amount function #generalizations

General ratings (exponential ratings)

definition

Is a totally ordered Abelian group and a (commutative) body, then is a mapping ${\displaystyle G}$ ${\displaystyle K}$

${\displaystyle v\colon K\to G\cup \{\infty \}}$

a non-Archimedean rating if the following properties are met:

• ${\displaystyle v(ab)=v(a)+v(b)}$
• ${\displaystyle v(a)=\infty \iff a=0}$
• ${\displaystyle v(a+b)\geq \min\{v(a),v(b)\}}$

for everyone . ${\displaystyle a,b\in K}$

${\displaystyle K}$then also means a valued body with a value group . ${\displaystyle v(K^{\times })\subseteq G}$

Two evaluations and are equivalent when true. Equivalence classes of ratings are also referred to as locations of a given body. ${\displaystyle v_{1}}$ ${\displaystyle v_{2}}$ ${\displaystyle v_{1}(a)<1\Longleftrightarrow v_{2}(a)<1}$

Ratings and rating rings

A health area is called a rating ring if it has the following properties: ${\displaystyle A}$

For each element of the quotient field of or applies . ${\displaystyle x}$ ${\displaystyle A}$ ${\displaystyle x\in A}$ ${\displaystyle x^{-1}\in A}$

If a valuation ring is a quotient body , you can define a valuation with a value group: ${\displaystyle A}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle G=K^{\times }/A^{\times }}$

${\displaystyle v\colon K\to G\cup \{\infty \},\quad v(x)=\left\{{\begin{matrix}\infty &x=0\\{}[x]&x\in K^{\times };\end{matrix}}\right.}$

where the image of in denotes ; the order on is defined by ${\displaystyle [x]}$ ${\displaystyle x}$ ${\displaystyle G=K^{\times }/A^{\times }}$ ${\displaystyle G}$

${\displaystyle [x]\geq [y]\iff xy^{-1}\in A}$ For ${\displaystyle x,y\in K^{\times }.}$

Conversely, if a valued body is valued , then it is ${\displaystyle K}$ ${\displaystyle v}$

${\displaystyle \{x\in K\mid v(x)\geq 0\}}$

an evaluation ring, which is then also called the evaluation ring for evaluation . The group is canonically isomorphic to the value group of . ${\displaystyle v}$ ${\displaystyle K^{\times }/A^{\times }}$ ${\displaystyle v}$

So for a body there is a bijective relationship between isomorphism classes of ratings and rating rings that are contained in. ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle K}$

Discreet reviews

definition

It is a body . Then is called a surjective function ${\displaystyle K}$

${\displaystyle v\colon K\to \mathbb {Z} \cup \{\infty \}}$

a discrete valuation , exponential valuation or non-Archimedean valuation if the following properties are met:

• ${\displaystyle v(ab)=v(a)+v(b)}$
• ${\displaystyle v(a)=\infty \iff a=0}$
• ${\displaystyle v(a+b)\geq \min\{v(a),v(b)\}}$

for everyone . together with is called discretely valued body. ${\displaystyle a,b\in K}$ ${\displaystyle K}$ ${\displaystyle v}$

Examples

• the rating on the rational numbers for a prime number ${\displaystyle p}$ ${\displaystyle p}$
• the zero or pole order of meromorphic functions at a fixed point

Discrete ratings and discrete rating rings

The subset

${\displaystyle A:=\left\{x\in K\mid v(x)\geq 0\right\}}$

forms a subring of , the evaluation ring of . It is a discrete evaluation ring with a maximum ideal , which is the main ideal . ${\displaystyle K}$ ${\displaystyle v}$ ${\displaystyle {\mathfrak {m}}:=\{x\mid x\in K,v(x)>0\}}$

Conversely, if a discrete evaluation ring is used, it is through ${\displaystyle (A,{\mathfrak {m}})}$

${\displaystyle v(x)=\sup \left\{k\in \mathbb {Z} \mid x\in {\mathfrak {m}}^{k}\right\}}$

a discrete valuation on the quotient field of defined. ${\displaystyle A}$

Discrete rating rings and discretely rated bodies correspond to one another.

p rating

Let it be a prime number . ${\displaystyle p}$

The -evaluation (also: the -adic evaluation or the -exponent) of a natural or whole number is the largest number , so that it is still divisible by . The rating indicates how often a prime number appears in the prime factorization of a natural or integer. ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle v_{p}(n)}$ ${\displaystyle n}$ ${\displaystyle k}$ ${\displaystyle n}$ ${\displaystyle p^{k}}$ ${\displaystyle p}$ ${\displaystyle p}$

Is

${\displaystyle n=p_{1}^{a_{1}}p_{2}^{a_{2}}...p_{k}^{a_{k}},}$

so is

${\displaystyle v_{p_{1}}(n)=a_{1},\quad v_{p_{2}}(n)=a_{2},\quad \ldots ,\quad v_{p_{k}}(n)=a_{k}.}$

If a prime number does not appear in the prime factorization of , then is . ${\displaystyle p}$ ${\displaystyle n}$ ${\displaystyle v_{p}(n)=0}$

You bet because every power of every prime divides 0 . ${\displaystyle v_{p}(0)=\infty }$

The evaluation of an integer is that of its amount . ${\displaystyle p}$

The rating of a rational number is the difference between the ratings of the numerator and the denominator: For a rational number with is ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle r={\tfrac {m}{n}}}$ ${\displaystyle m,n\in \mathbb {Z} }$

${\displaystyle v_{p}(r)=v_{p}(m)-v_{p}(n).}$

If p only appears in the denominator of the ( fully abbreviated ) fraction , it is therefore a negative number . ${\displaystyle m/n}$ ${\displaystyle v_{p}(r)}$

The evaluation of rational numbers plays an important role in one type of construction of the p-adic numbers : the function ${\displaystyle p}$

${\displaystyle r\mapsto p^{-v_{p}(r)}}$

forms a non-Archimedean amount on the rational numbers .

p -integer and S -integer

An -integer (also " -adic integer" or "for integer") is a rational number that has a nonnegative evaluation ; H. in which the denominator cannot be divided by in a fully abbreviated fraction representation . Rational numbers that are not -integer are sometimes called " -from". ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p}$

The set of all -integer numbers is a subring of that is written. is a discrete evaluation ring , in particular there is exactly one irreducible element apart from associated ones , namely . ${\displaystyle p}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Z} _{(p)}}$ ${\displaystyle \mathbb {Z} _{(p)}}$ ${\displaystyle p}$

If, more generally, is a set of prime numbers, then an -integer is a rational number that is -integral for each (!), I.e. H. in which the denominator can only be divided by prime numbers in a completely abbreviated fraction representation . The set of -integer numbers forms a subring of . ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle p}$ ${\displaystyle p\notin S}$ ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle \mathbb {Z} _{S}}$ ${\displaystyle \mathbb {Q} }$

Examples

• For is . ${\displaystyle S=\emptyset }$ ${\displaystyle \mathbb {Z} _{S}=\mathbb {Z} }$
• For a prime number and is , the discrete valuation ring of the -integer numbers. ${\displaystyle p\in \mathbb {P} }$ ${\displaystyle S=\mathbb {P} \setminus \{p\}}$ ${\displaystyle \mathbb {Z} _{S}=\mathbb {Z} _{(p)}}$ ${\displaystyle p}$
• For is the ring of terminating decimal fractions (which can be represented by a finite sequence of digits) . ${\displaystyle S=\{2,5\}}$ ${\displaystyle \mathbb {Z} _{S}}$

Generalizations

The concept of a norm can be understood more generally by allowing arbitrary vector spaces over evaluated bodies , i.e. bodies with an absolute value , instead of vector spaces over the body of real or complex numbers . Another generalization is that the vector space is replaced by a - (left) - module over a unitary ring of magnitude . A function is then called a norm on the module if the three norm properties definiteness, absolute homogeneity and subadditivity are fulfilled for all and all scalars . When the base ring of the amount by a pseudo amount is replaced in the module and the homogeneity is attenuated to Subhomogenität, one obtains a pseudo standard . ${\displaystyle \mathbb {K} }$ ${\displaystyle (K,|\cdot |)}$ ${\displaystyle |\cdot |}$ ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle (R,|\cdot |)}$ ${\displaystyle \|\cdot \|\colon M\to \mathbb {R} _{+}}$ ${\displaystyle M}$ ${\displaystyle x,y\in M}$ ${\displaystyle \alpha \in R}$ ${\displaystyle R}$ ${\displaystyle M}$

literature

• BL van der Waerden : Algebra II , Springer-Verlag (1967), ISBN 3-540-03869-8 , Chapter eighteenth: "Evaluated bodies", pp. 200-234.
• J. Neukirch : Algebraic Number Theory , Springer-Verlag (2006), ISBN 3-5403-7547-3 , Chapter II: "Evaluation Theory", pp. 103-191.
• Serge Lang : Algebra , Springer (2005), ISBN 0-387-95385-X , Absolute Values, pp. 465-499.

Web links

• Valuation (MathWorld)
• Valuation (Encyclopedia of Mathematics)

Individual evidence

1. ↑ Waerden, op.cit., P. 200
2. ↑ Neukirch, op.cit., P. 121
3. ^ Heinz-Dieter Ebbinghaus et al .: Numbers. 2nd edition, Springer, Berlin / Heidelberg 1988, chapter 4, p. 65
4. ↑ Falko Lorenz: Introduction to Algebra II . 2nd Edition. Spectrum Academic Publishing House, 1997, p. 69 .