Absolute value (algebra)

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: 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, 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 d ( f , g ) = | f − g | {displaystyle d(f,g)=|f-g|} 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 | 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. 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, any rational number q can be written as pn(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