alternative definition of valuation
A valuation on a field is a map such that
if an only if
The quantity which appears in the third criterion is a positive real number which is known as the .
There is some flexibility in the choice of the constant in this definition — one can replace by a larger number . To deal with this ambiguity, one defines the of the valuation as
so a valuation in the old sense is a valuation in the new sense with constant 2.
On the other hand, suppose that satisfies the alternative definition with constant . Then we have the following result.
The proof of this assertion is given in a supplement to this entry.
The foregoing discussion shows that the new definition is more general than the old definition precisely when . However, this extra generalty is not as great as it might seem at first sight. As is obvious from examining the definition, if is a valuation, then so is for any power . Furthermore, if the valuation has constant , then valuation has constant . Therefore, given any valuation in the sense of this entry, there will exist a number such that is a valuation in the sense of the parent entry. Moreover, given the fact that two valuations which are powers of each other are equivalent, one sees that the extra generality is not that interesting since the new valuations are equivalent to the old valuations.
|Title||alternative definition of valuation|
|Date of creation||2013-03-22 14:55:47|
|Last modified on||2013-03-22 14:55:47|
|Last modified by||rspuzio (6075)|