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
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![{\ displaystyle \ varphi \ colon K \ to P}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2c7f40e157f209ecd9d75c7857abd7e8b2bda0)
![P](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a)
-
and
![{\ displaystyle \ varphi (xy) = \ varphi (x) \ varphi (y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12a01cd413bae6c388f2f2f6b0947640d0c4f901)
![{\ displaystyle \ varphi (x + y) \ leq \ varphi (x) + \ varphi (y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9dd59f45ba8d9f8295518be43023f73b86e802eb)
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.
![| x |](https://wikimedia.org/api/rest_v1/media/math/render/svg/4eb41e5fd5dc37eaa1718dfbf4bc082edb991936)
![{\ displaystyle | \ cdot | \ colon \ mathbb {C} \ to \ mathbb {R}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92d4e0e1790290c0bbf2bffcea37260ae3d6d917)
![{\ displaystyle | \ cdot | \ colon K \ to \ mathbb {R}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99da23743bd6d9226f74c32634e68542b02ee64f)
![{\ displaystyle | n | \ leq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/afb202a67ffa25165515d2df775a36eacad7867a)
![n \ in \ mathbb {N}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b)
General ratings (exponential ratings)
definition
Is a totally ordered Abelian group and a (commutative) body, then is a mapping
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![v \ colon K \ to G \ cup \ {\ infty \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8715ee08f8e2404f53092931305994ccf56ed387)
a non-Archimedean rating if the following properties are met:
![v (ab) = v (a) + v (b)](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc0146408cb71591e419dfb56b338ed5a166ecbb)
![v (a) = \ infty \ iff a = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a44324c7d9a0026b50f2d81d13599aaef275db7)
![v (a + b) \ geq \ min \ {v (a), v (b) \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78618a0631bbe51fa084fd695716f82c384e773d)
for everyone .
![a, b \ in K](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ff3a49d65fc590e33a74fd613900dd5924d6ca)
then also means a valued body with a value group .
![v (K ^ {\ times}) \ subseteq G](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b7b835ff0d56a27691e7c6d02ec2a657880382c)
Two evaluations and are equivalent when true. Equivalence classes of ratings are also referred to as locations of a given body.
![v_ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98d33f5d498d528bd8c10edc8ac8c34347f32b3a)
![v_ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb04c423c2cb809c30cac725befa14ffbf4c85f3)
![{\ displaystyle v_ {1} (a) <1 \ Longleftrightarrow v_ {2} (a) <1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31d9046f8fc7c6e70d5fa1eb4700bceaca130190)
Ratings and rating rings
A health area is called a rating ring if it has the following properties:
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
- For each element of the quotient field of or applies .
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![x \ in A](https://wikimedia.org/api/rest_v1/media/math/render/svg/27bcc9b2afb295d4234bc294860cd0c63bcad2ca)
![x ^ {{- 1}} \ in A](https://wikimedia.org/api/rest_v1/media/math/render/svg/661765af3e2f411a8d6b05f9832a795846561f03)
If a valuation ring is a quotient body , you can define a valuation with a value group:
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![G = K ^ {\ times} / A ^ {\ times}](https://wikimedia.org/api/rest_v1/media/math/render/svg/026d9463c4880978a673675266b66572918c6680)
![v \ colon K \ to G \ cup \ {\ infty \}, \ quad v (x) = \ left \ {{\ begin {matrix} \ infty & x = 0 \\ {} [x] & x \ in K ^ {\ times}; \ end {matrix}} \ right.](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe90feb9c73794b4919566031052d2fb2e30c83e)
where the image of in denotes ; the order on is defined by
![[x]](https://wikimedia.org/api/rest_v1/media/math/render/svg/07548563c21e128890501e14eb7c80ee2d6fda4d)
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
![G = K ^ {\ times} / A ^ {\ times}](https://wikimedia.org/api/rest_v1/media/math/render/svg/026d9463c4880978a673675266b66572918c6680)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
-
For
Conversely, if a valued body is valued , then it is
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![v](https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597)
![\ {x \ in K \ mid v (x) \ geq 0 \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c8c28ac54326ad900f554b95c5a53adb1362e1b)
an evaluation ring, which is then also called the evaluation ring for evaluation . The group is canonically isomorphic to the value group of .
![v](https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597)
![K ^ {\ times} / A ^ {\ times}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b34f76776ca369a096c2e6fc2c1f89c4f996310c)
![v](https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597)
So for a body there is a bijective relationship between isomorphism classes of ratings and rating rings that are contained in.
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
Discreet reviews
definition
It is a body . Then is called a surjective function
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![v \ colon K \ to {\ mathbb Z} \ cup \ {\ infty \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c7e4c0c4f4df889c3a02076ec5af29fe95319ac)
a discrete valuation , exponential valuation or non-Archimedean valuation if the following properties are met:
![v (ab) = v (a) + v (b)](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc0146408cb71591e419dfb56b338ed5a166ecbb)
![v (a) = \ infty \ iff a = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a44324c7d9a0026b50f2d81d13599aaef275db7)
![v (a + b) \ geq \ min \ {v (a), v (b) \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78618a0631bbe51fa084fd695716f82c384e773d)
for everyone . together with is called discretely valued body.
![a, b \ in K](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ff3a49d65fc590e33a74fd613900dd5924d6ca)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![v](https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597)
Examples
- the rating on the rational numbers for a prime number
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
- the zero or pole order of meromorphic functions at a fixed point
Discrete ratings and discrete rating rings
The subset
![A: = \ left \ {x \ in K \ mid v (x) \ geq 0 \ right \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/878d0098fe5c9278623490f49acbcd3adb4ea57f)
forms a subring of , the evaluation ring of . It is a discrete evaluation ring with a maximum ideal , which is the main ideal .
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![{\ mathfrak m}: = \ {x \ mid x \ in K, v (x)> 0 \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86a9cb4048210f1bfc3770bae87212a622a3d64a)
Conversely, if a discrete evaluation ring is used, it is through
![(A, {\ mathfrak m})](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e31fe49c6f1961f10596bec1bc271e67532e1a6)
![v (x) = \ sup \ left \ {k \ in {\ mathbb Z} \ mid x \ in {\ mathfrak m} ^ {k} \ right \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce62eace03258baebdcd54e12c399ebd04c863b8)
a discrete valuation on the quotient field of defined.
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
Discrete rating rings and discretely rated bodies correspond to one another.
p rating
Let it be a prime number .
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
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.
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![v_ {p} (n)](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab7ef07cef84d44d3fd868891d304b7afafae1b8)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![p ^ {k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e017c102135ab13bdf501dc1c1b5fd1840a97822)
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
Is
![n = p_ {1} ^ {{a_ {1}}} p_ {2} ^ {{a_ {2}}} ... p_ {k} ^ {{a_ {k}}},](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c2965c9019d2d247794d2a2bba2d822c533e3a6)
so is
![v _ {{p_ {1}}} (n) = a_ {1}, \ quad v _ {{p_ {2}}} (n) = a_ {2}, \ quad \ ldots, \ quad v _ {{p_ { k}}} (n) = a_ {k}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/6838ca0dbe5fc9e10a8ea9b870a44ffd884fc058)
If a prime number does not appear in the prime factorization of , then is .
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![v_ {p} (n) = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/219116cea02089e89cb61dd026a6f76afe2954a9)
You bet because every power of every prime divides 0 .
![v_ {p} (0) = \ infty](https://wikimedia.org/api/rest_v1/media/math/render/svg/31dc032f45b2f3cca53885fa12e613be7e129e95)
The evaluation of an integer is that of its amount .
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
The rating of a rational number is the difference between the ratings of the numerator and the denominator: For a rational number with is
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![{\ displaystyle r = {\ tfrac {m} {n}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09e73d1beb88bd72dc3b5dc738e183eeaf05af94)
![m, n \ in {\ mathbb Z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f932c786e686277b8f060aae74134ce9fc3e3348)
![v_ {p} (r) = v_ {p} (m) -v_ {p} (n).](https://wikimedia.org/api/rest_v1/media/math/render/svg/9df00076975dc55355ea3b415252cb6a3699bd84)
If p only appears in the denominator of the ( fully abbreviated ) fraction , it is therefore a negative number .
![m / n](https://wikimedia.org/api/rest_v1/media/math/render/svg/2eebcb27a9df80445dbe86eefee5d131d6e0e7e8)
![v_ {p} (r)](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e0b37c9fda28046df84d0784dff3585e5e2039f)
The evaluation of rational numbers plays an important role in one type of construction of the p-adic numbers : the function
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![r \ mapsto p ^ {{- v_ {p} (r)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c5b9f1e7e7eb736234b5adad2613a97f6058180)
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".
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
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 .
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![\ mathbb {Q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a)
![{\ mathbb Z} _ {{(p)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31ece65379ee3430084adc10c17f0b4d0a0fad16)
![{\ mathbb Z} _ {{(p)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31ece65379ee3430084adc10c17f0b4d0a0fad16)
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
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 .
![S.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
![S.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![p \ notin S.](https://wikimedia.org/api/rest_v1/media/math/render/svg/688846d6ed99d68416589a179ba3044e6f854348)
![S.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
![S.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
![{\ mathbb Z} _ {S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06accfafc512e72329e0a582e1dc513319730a74)
![\ mathbb {Q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a)
- Examples
- For is .
![{\ displaystyle S = \ emptyset}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a7c38be46a9b3fe15e5893edc24def5ba893e5e)
![{\ displaystyle \ mathbb {Z} _ {S} = \ mathbb {Z}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a69fafabe52a71bfb1f00024e8cbcf950e62f04)
- For a prime number and is , the discrete valuation ring of the -integer numbers.
![{\ displaystyle p \ in \ mathbb {P}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7891a37dcb8b0ee507f9ef2038a853e245d76657)
![{\ displaystyle S = \ mathbb {P} \ setminus \ {p \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/149fa5cb66c79b154405822b2a0db1e0e32a2627)
![{\ displaystyle \ mathbb {Z} _ {S} = \ mathbb {Z} _ {(p)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3baa0ab3a1f37b186ff456f116ad3500d684d30e)
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
- For is the ring of terminating decimal fractions (which can be represented by a finite sequence of digits) .
![{\ displaystyle S = \ {2.5 \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f0d4c858977aabf401f1ce5483a4dbfc708df7b)
![{\ displaystyle \ mathbb {Z} _ {S}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06accfafc512e72329e0a582e1dc513319730a74)
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 .
![\ mathbb {K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1848c435e64864e9ad4efa7e46bd6bc900c35c99)
![| \ cdot |](https://wikimedia.org/api/rest_v1/media/math/render/svg/4570d0a1c9fb8f2f413f0b73ce846dd1eb1dca3f)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![(R, | \ cdot |)](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f73aadb9d6be91b85a81d8b382046ebbe109c5b)
![\ | \ cdot \ | \ colon M \ to \ mathbb {R} _ {+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d032ab822eaf47d8d22696557f8f6144ff618f10)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![x, y \ in M](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea304ca242a255b620d3dd16ec47f19efc2e7ab8)
![\ alpha \ in R](https://wikimedia.org/api/rest_v1/media/math/render/svg/b09907bd4a5d9f895bddca8cc8d829c3e214b5e8)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
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 .