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Concept
Absolute value (algebra)
Summary
In algebra, 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. More precisely, if D is an integral domain, then an absolute value is any mapping |x| from D to the real numbers R satisfying:
It follows from these axioms that |1| = 1 and |-1| = 1. Furthermore, for every positive integer n,
|n| = |1 + 1 + ... + 1 (n times)| = |−1 − 1 − ... − 1 (n times)| ≤ n.
The classical "absolute value" is one in which, for example, |2|=2, but many other functions fulfill the requirements stated above, for instance the square root of the classical absolute value (but not the square thereof).
An absolute value induces a metric (and thus a topology) by
The standard absolute value on the integers.
The standard absolute value on the complex numbers.
The p-adic absolute value on the rational numbers.
If R is the field of rational functions 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
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 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 or non-Archimedean absolute value, and otherwise an Archimedean absolute value.
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|1e = |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.
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