Rational numbers are the quotient field of integers.
2.
Rational expressions are the quotient field of the polynomials ( over some integral domain ).
3.
The expression " quotient field " may sometimes run the risk of confusion with the quotient of a ring by an ideal, which is a quite different concept.
4.
They were introduced by for abelian varieties over the quotient field of a Dedekind domain " R " with perfect residue fields, and extended this construction to semiabelian varieties over all Dedekind domains.
5.
Quotient rings are distinct from the so-called'quotient field', or field of fractions, of an integral domain as well as from the more general'rings of quotients'obtained by localization.
6.
If " A " is a Dedekind domain whose quotient field is an algebraic number field ( a finite extension of the rationals ) then shows that SK 1 ( " A " ) vanishes.
7.
:: I had the same thought-although I think that is a basis over the quotient field of K, not over K itself ( but that is probably what the questioner meant anyway ) . talk ) 16 : 54, 3 December 2008 ( UTC)
8.
On top of this may be attached any number of symbolic variables t _ 1, t _ 2, \ dots, t _ n, thereby creating the polynomial ring F [ t _ 1, t _ 2, \ dots, t _ n ] and its quotient field.
9.
On the usual local fields ( typically completions of number fields or the quotient fields of local rings of algebraic curves ) there is a unique surjective discrete valuation ( of rank 1 ) associated to a choice of a local parameter of the fields, unless they are archimedean local fields such as the real numbers and complex numbers.
10.
Conversely, the valuation \ nu : A \ rightarrow \ Z \ cup \ { \ infty \ } on a discrete valuation ring A can be extended in a unique way to a discrete valuation on the quotient field K = \ text { Quot } ( A ); the associated discrete valuation ring \ mathcal { O } _ K is just A.
मोबाइल