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
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and
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.
General ratings (exponential ratings)
definition
Is a totally ordered Abelian group and a (commutative) body, then is a mapping
a non-Archimedean rating if the following properties are met:
for everyone .
then also means a valued body with a value group .
Two evaluations and are equivalent when true. Equivalence classes of ratings are also referred to as locations of a given body.
Ratings and rating rings
A health area is called a rating ring if it has the following properties:
- For each element of the quotient field of or applies .
If a valuation ring is a quotient body , you can define a valuation with a value group:
where the image of in denotes ; the order on is defined by
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For
Conversely, if a valued body is valued , then it is
an evaluation ring, which is then also called the evaluation ring for evaluation . The group is canonically isomorphic to the value group of .
So for a body there is a bijective relationship between isomorphism classes of ratings and rating rings that are contained in.
Discreet reviews
definition
It is a body . Then is called a surjective function
a discrete valuation , exponential valuation or non-Archimedean valuation if the following properties are met:
for everyone . together with is called discretely valued body.
Examples
- the rating on the rational numbers for a prime number
- the zero or pole order of meromorphic functions at a fixed point
Discrete ratings and discrete rating rings
The subset
forms a subring of , the evaluation ring of . It is a discrete evaluation ring with a maximum ideal , which is the main ideal .
Conversely, if a discrete evaluation ring is used, it is through
a discrete valuation on the quotient field of defined.
Discrete rating rings and discretely rated bodies correspond to one another.
p rating
Let it be a prime number .
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.
Is
so is
If a prime number does not appear in the prime factorization of , then is .
You bet because every power of every prime divides 0 .
The evaluation of an integer is that of its amount .
The rating of a rational number is the difference between the ratings of the numerator and the denominator: For a rational number with is
If p only appears in the denominator of the ( fully abbreviated ) fraction , it is therefore a negative number .
The evaluation of rational numbers plays an important role in one type of construction of the p-adic numbers : the function
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".
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 .
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 .
- Examples
- For is .
- For a prime number and is , the discrete valuation ring of the -integer numbers.
- For is the ring of terminating decimal fractions (which can be represented by a finite sequence of digits) .
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 .
literature
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BL van der Waerden : Algebra II , Springer-Verlag (1967), ISBN 3-540-03869-8 , Chapter eighteenth: "Evaluated bodies", pp. 200-234.
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J. Neukirch : Algebraic Number Theory , Springer-Verlag (2006), ISBN 3-5403-7547-3 , Chapter II: "Evaluation Theory", pp. 103-191.
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Serge Lang : Algebra , Springer (2005), ISBN 0-387-95385-X , Absolute Values, pp. 465-499.
Web links
Individual evidence
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↑ Waerden, op.cit., P. 200
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↑ Neukirch, op.cit., P. 121
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^ Heinz-Dieter Ebbinghaus et al .: Numbers. 2nd edition, Springer, Berlin / Heidelberg 1988, chapter 4, p. 65
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↑ Falko Lorenz: Introduction to Algebra II . 2nd Edition. Spectrum Academic Publishing House, 1997, p. 69 .