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)
Famous quotes containing the word places:
“All men are lonely. But sometimes it seems to me that we Americans are the loneliest of all. Our hunger for foreign places and new ways has been with us almost like a national disease. Our literature is stamped with a quality of longing and unrest, and our writers have been great wanderers.”
—Carson McCullers (1917–1967)
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—Henry David Thoreau (1817–1862)
“The sense of place is unavoidable in western Dakota, and maybe that’s our gift to the world. ...In these places you wait, and the places mold you.”
—Kathleen Norris (b. 1947)