In
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
, an absolute value (also called a valuation, magnitude, or norm, although "
norm" usually refers to a specific kind of absolute value on a
field) is a
function which measures the "size" of elements in a field or
integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
. More precisely, if ''D'' is an integral domain, then an absolute value is any mapping , x, from ''D'' to the
real numbers
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every re ...
R satisfying:
It follows from these axioms that , 1, = 1 and , -1, = 1. Furthermore, for every positive
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
''n'',
:, ''n'', = , 1 + 1 + ... + 1 (''n'' times), = , −1 − 1 − ... − 1 (''n'' times), ≤ ''n''.
The classical "
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
" is one in which, for example, , 2, =2, but many other functions fulfill the requirements stated above, for instance the
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
...
of the classical absolute value (but not the square thereof).
An absolute value induces a
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathe ...
(and thus a
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
) by
Examples
*The standard absolute value on the integers.
*The standard absolute value on the
complex numbers.
*The
''p''-adic absolute value on the
rational numbers
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
.
*If ''R'' is the field of
rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
s over a field ''F'' and
is a fixed
irreducible element of ''R'', then the following defines an absolute value on ''R'': for
in ''R'' define
to be
, where
and
Types of absolute value
The trivial absolute value is the absolute value with , ''x'', =0 when ''x''=0 and , ''x'', =1 otherwise. Every integral domain can carry at least the trivial absolute value. The trivial value is the only possible absolute value on a
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
because any non-zero element can be raised to some power to yield 1.
If an absolute value satisfies the stronger property , ''x'' + ''y'', ≤ max(, ''x'', , , ''y'', ) for all ''x'' and ''y'', then , ''x'', is called an
ultrametric
In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to d(x,z)\leq\max\left\. Sometimes the associated metric is also called a non-Archimedean metric or super-metric. Although some of the theorems ...
or non-Archimedean absolute value, and otherwise an Archimedean absolute value.
Places
If , ''x'',
1 and , ''x'',
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 for all ''x''. If two nontrivial absolute values are equivalent, then for some exponent ''e'' we have , ''x'',
1''e'' = , ''x'',
2 for all ''x''. Raising an absolute value to a power less than 1 results in another absolute value, but raising to a power greater than 1 does not necessarily result in an absolute value. (For instance, squaring the usual absolute value on the real numbers yields a function which is not an absolute value because it violates the rule , ''x''+''y'', ≤ , ''x'', +, ''y'', .) Absolute values up to equivalence, or in other words, an
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
of absolute values, is called a
place
Place may refer to:
Geography
* Place (United States Census Bureau), defined as any concentration of population
** Census-designated place, a populated area lacking its own municipal government
* "Place", a type of street or road name
** O ...
.
Ostrowski's theorem states that the nontrivial places of the
rational numbers
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
Q are the ordinary
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
and the
''p''-adic absolute value for each prime ''p''. For a given prime ''p'', any rational number ''q'' can be written as ''p''
''n''(''a''/''b''), where ''a'' and ''b'' are integers not divisible by ''p'' and ''n'' is an integer. The ''p''-adic absolute value of ''q'' is
:
Since the ordinary absolute value and the ''p''-adic absolute values are absolute values according to the definition above, these define places.
Valuations
If for some ultrametric absolute value and any base ''b'' > 1, we define ''ν''(''x'') = −log
''b'', ''x'', for ''x'' ≠ 0 and ''ν''(0) = ∞, where ∞ is ordered to be greater than all real numbers, then we obtain a function from ''D'' to R ∪ , with the following properties:
* ''ν''(''x'') = ∞ ⇒ ''x'' = 0,
* ''ν''(''xy'') = ''ν''(''x'')+''ν''(''y''),
* ''ν''(''x'' + ''y'') ≥ min(ν(''x''), ''ν''(''y'')).
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''.
Completions
Given an integral domain ''D'' with an absolute value, we can define the
Cauchy sequence
In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
s of elements of ''D'' with respect to the absolute value by requiring that for every ε > 0 there is a positive integer ''N'' such that for all integers ''m'', ''n'' > ''N'' one has , ''x''
''m'' − ''x''
''n'', < ε. Cauchy sequences form a
ring under pointwise addition and multiplication. One can also define null sequences as sequences (''a''
''n'') of elements of ''D'' such that , ''a''
''n'', converges to zero. Null sequences are a
prime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together wi ...
in the ring of Cauchy sequences, and the
quotient ring
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
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 , ''x'', .
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
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals c ...
, 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.
Another theorem of
Alexander Ostrowski
Alexander Markowich Ostrowski ( uk, Олександр Маркович Островський; russian: Алекса́ндр Ма́ркович Остро́вский; 25 September 1893, in Kiev, Russian Empire – 20 November 1986, in Mont ...
has it that any field complete with respect to an
Archimedean absolute value is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to either the real or the complex numbers, and the valuation is equivalent to the usual one. The Gelfand-Tornheim theorem states that any field with an Archimedean valuation is isomorphic to a
subfield of C, the valuation being equivalent to the usual absolute value on C.
Fields and integral domains
If ''D'' is an integral domain with absolute value , ''x'', , then we may extend the definition of the absolute value to the
field of fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
of ''D'' by setting
:
On the other hand, if ''F'' is a field with ultrametric absolute value , ''x'', , then the set of elements of ''F'' such that , ''x'', ≤ 1 defines a
valuation ring In abstract algebra, a valuation ring is an integral domain ''D'' such that for every element ''x'' of its field of fractions ''F'', at least one of ''x'' or ''x''−1 belongs to ''D''.
Given a field ''F'', if ''D'' is a subring of ''F'' suc ...
, which is a
subring
In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those ...
''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 divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right ze ...
s and is an integral domain. It has a unique
maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals c ...
consisting of all ''x'' such that , ''x'', < 1, and is therefore a
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebrai ...
.
Notes
References
*
*
* Chapter 9, paragraph 1 "''Absolute values''".
*
{{refend
Abstract algebra