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 of childhood’s unanswered questions must finally be passed back to the town and answered there. Heroes and bogey men, values and dislikes, are first encountered and labeled in that early environment. In later years they change faces, places and maybe races, tactics, intensities and goals, but beneath those penetrable masks they wear forever the stocking-capped faces of childhood.”
—Maya Angelou (b. 1928)
“In many places the road was in that condition called repaired, having just been whittled into the required semicylindrical form with the shovel and scraper, with all the softest inequalities in the middle, like a hog’s back with the bristles up.”
—Henry David Thoreau (1817–1862)
“I was never one to patiently pick up broken fragments and glue them together again and tell myself that the mended whole was as good as new. What is broken is broken—and I’d rather remember it as it was at its best than mend it and see the broken places as long as I lived.... I wish I could care what you do or where you go, but I can’t. My dear, I don’t give a damn.”
—Margaret Mitchell (1900–1949)