In mathematics, an absolute value is a function which measures the "size" of elements in a field or integral domain. More precisely, if D is an integral domain, then an absolute value is any mapping | | from D to the real numbers R satisfying Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠ 1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ... Please refer to Real vs. ...

• |x| ≥ 0
• |x| = 0 if and only if x = 0
• |xy| = |x||y|
• |x+y| ≤ |x|+|y|

Note that some authors use the term valuation instead of "absolute value". Model Theory In logic and model theory, a valuation is a map from the set of variables of a first-order language to the universe of some interpretation of that language. ...

Types of absolute value

If |x+y| satisfies the stronger property

|x+y| ≤ max(|x|, |y|),

then | | is called an ultrametric or non-Archimedean absolute value, and otherwise an Archimedean absolute value. In mathematics, an ultrametric space is a special kind of metric space. ...

If |x|=1 for all nonzero values, the absolute value is called trivial and otherwise nontrivial.

Places

If | |₁ and | |₂ are two absolute values on the same integral domain D, then the two absolute values are equivalent if |x|₁ < 1 if and only if |x|₂ < 1. If two nontrivial absolute values are equivalent, then for some exponent e, we have |x|₁^e = |x|₂. Absolute values up to equivalence, or in other words, an equivalence class of absolute values, is called a place.

Ostrowski's theorem states that the nontrivial places of the rational numbers Q are the ordinary absolute value and the p-adic absolute value for each prime p. For a given prime p, the p-adic absolute value of the rational number , where a and b are integers not divisible by p, is . Since the ordinary absolute value and the p-adic absolute values are normalized, these define places. Ostrowskis theorem, due to Alexander Ostrowski, states that any non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a p-adic absolute value. ... In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ... In mathematics, the p-adic number systems were first described by Kurt Hensel in 1897. ...

Valuations

If for some ultrametric absolute value we define ν(x) = − log_b( | x | ) for any base b>1, and add to this the special value ν(0)=∞, which is ordered to be greater than all real numbers, we obtain a function from D to R ∪ ∞, with the following properties:

Such a function is known as a valuation in the terminology of Bourbaki, but other authors use the term valuation for absolute value and then say exponential valuation instead of valuation. Model Theory In logic and model theory, a valuation is a map from the set of variables of a first-order language to the universe of some interpretation of that language. ... Nicolas Bourbaki is the pseudonym under which a group of mainly French 20th-century mathematicians wrote a series of books of exposition of modern advanced mathematics, beginning in 1935. ...

Completions

Given an integral domain D with an absolute value, we can define the Cauchy sequences of elements of D with respect to the absolute value by requiring that for every r > 0 there is a positive integer N such that for all integers m, n > N one has | x_m − x_n | < r. It is not hard to show that Cauchy sequences under pointwise addition and multiplication form a ring. One can also define null sequences as sequences of elements of D such that |a_n| converges to zero. Null sequences are a prime ideal in the ring of Cauchy sequences, and the quotient ring is therefore an integral domain. The domain D is embedded in this quotient ring, called the completion of D with respect to the absolute value | |. In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For...

Since fields are integral domains, this is also a construction for the completion of a field with respect to an absolute value. To show that the result is a field, and not just an integral domain, we can either show that null sequences form a maximal ideal, or else construct the inverse directly. The latter can be easily done by taking, for all nonzero elements of the quotient ring, a sequence starting from a point beyond the last zero element of the sequence. Any nonzero element of the quotient ring will differ by a null sequence from such a sequence, and by taking pointwise inversion we can find a representative inverse element. In mathematics, more specifically in ring theory a maximal ideal is an ideal which is maximal (with respect to set inclusion) amongst all proper ideals, i. ...

Another theorem of Alexander Ostrowski has it that any field complete with respect to the usual Archimedean absolute value is isomorphic to either the real or the complex numbers. Ultrametric complete fields are far more numerous, however. Alexander Markowich Ostrowski (25 September 1893, Kiev, Ukraine - 20 Nov 1986, Montagnola, Lugano, Switzerland), was a mathematician. ... Archimedean refers to a thing named after the Greek mathematician Archimedes. ...

Fields and integral domains

If D is an integral domain with absolute value | |, then we may extend the definition of the absolute value to the field of fractions of D by setting In mathematics, every integral domain can be embedded in a field; the smallest field which can be used is the field of fractions of the integral domain. ...

On the other hand, if F is a field with ultrametric absolute value | |, then the set of elements of F such that |x| ≤ 1 defines a valuation ring, which is a subring D of F such that for every nonzero element x of F, at least one of x or x^-1 belongs to D. Since F is a field, D has no zero divisors and is an integral domain. It has a unique maximal ideal consisting of all x such that |x|<1, and is therefore a local ring. In abstract algebra, local rings are certain rings that are comparatively simple and serve to describe the local behavior of functions defined on varieties or manifolds. ... In mathematics, more specifically in ring theory a maximal ideal is an ideal which is maximal (with respect to set inclusion) amongst all proper ideals, i. ... In mathematics, more particularly in abstract algebra, local rings are certain rings that are comparatively simple, and serve to describe the local behavior of functions defined on varieties or manifolds. ...

See also

• Ostrowski's theorem
• Valuation
• Absolute value
• Valuation ring
• Local ring

Ostrowskis theorem, due to Alexander Ostrowski, states that any non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a p-adic absolute value. ... Model Theory In logic and model theory, a valuation is a map from the set of variables of a first-order language to the universe of some interpretation of that language. ... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ... In abstract algebra, local rings are certain rings that are comparatively simple and serve to describe the local behavior of functions defined on varieties or manifolds. ... In mathematics, more particularly in abstract algebra, local rings are certain rings that are comparatively simple, and serve to describe the local behavior of functions defined on varieties or manifolds. ...

References

• Nicolas Bourbaki (1972). Commutative Algebra. Addison-Wesley. 
• Gerald J. Janusz (1996, 1997). Algebraic Number Fields, 2^nd edition, American Mathematical Society. ISBN 0-8218-0429-4. 
• Nathan Jacobson (1989). Basic algebra II, 2^nd ed., W H Freeman. ISBN 0-7167-1933-9.  Chapter 9, paragraph 1 "Absolute values".