Absolute value (algebra)
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 | 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,
Note that some authors use the terms valuation, norm, or magnitude instead of "absolute value". However, the word "norm" usually refers to a specific kind of absolute value on a field (and which is also applied to other vector spaces).
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).
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 element can be raised to some power to yield 1.
If | x + y | satisfies the stronger property | x + y | ≤ max(|x|, |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. If two nontrivial absolute values are equivalent, then for some exponent e, we have | x |1e = | x |2. 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 would violate the rule |x + y| ≤ |x| + |y|.) Absolute values up to equivalence, or in other words, an equivalence class of absolute values, is called a place.
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