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. If two nontrivial absolute values are equivalent, then for some exponent e, we have | x |1e = | x |2. (Note that we cannot necessarily raise one absolute value to some power and obtain another. For instance, squaring the usual absolute value on the real numbers yields a function which is not an absolute value.) 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 q = pn(a/b), 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.
Read more about this topic: Absolute Value (algebra)
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