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 places are distant from heaven alike.”
—Robert Burton (1577–1640)
“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)
“The mother whose self-image is dependent on her children places on those children the responsibility for her own identity, and her involvement in the details of their lives can put great pressure on the children. A child suffers when everything he or she does is extremely important to a parent; this kind of over-involvement can turn even a small problem into a crisis.”
—Grace Baruch (20th century)