Remark. The coefficients in a polynomial function need not be unique; e.g. if is the ring (and field) of two elements, then the polynomials and both may be used for the same polynomial function. However, if we stipulate that is an infinite integral domain, the coefficients are guaranteed to be unique.
The set of all polynomial functions of , being a subset of the set of all functions from to , is here denoted by .
If is a commutative ring, then the set of all polynomial functions of , equipped with the operations
is a commutative ring.
Proof. It’s straightforward to show that the function set forms a commutative ring when equipped with the operations “” and “” defined as (1). We show now that forms a subring of . Let and be any two polynomial functions given by
Then we can give by
where and (resp. ) for (resp. ). This means that . Secondly, the equation
signifies that . Because also the function given by
and satisfying belongs to , the subset is a subring of .
|Date of creation||2013-03-22 15:40:34|
|Last modified on||2013-03-22 15:40:34|
|Last modified by||pahio (2872)|
|Synonym||ring of polynomial functions|