1 Polynomial rings in one variable
Let be a ring. The polynomial ring over in one variable is the set of all sequences in with only finitely many nonzero terms. If is an element in , with for all , then we usually write this element as
Elements of are called polynomials in the indeterminate with coefficients in . The ring elements are called coefficients of the polynomial, and the degree of a polynomial is the largest natural number for which , if such an exists. When a polynomial has all of its coefficients equal to , its degree is usually considered to be undefined, although some people adopt the convention that its degree is .
A monomial is a polynomial with exactly one nonzero coefficient. Similarly, a binomial is a polynomial with exactly two nonzero coefficients, and a trinomial is a polynomial with exactly three nonzero coefficients.
Addition and multiplication of polynomials is defined by
is a –graded ring under these operations, with the monomials of degree exactly comprising the graded component of . The zero element of is the polynomial whose coefficients are all , and when has a multiplicative identity , the polynomial whose coefficients are all except for is a multiplicative identity for the polynomial ring .
2 Polynomial rings in finitely many variables
The polynomial ring over in two variables is defined to be . Elements of are called polynomials in the indeterminates and with coefficients in . A monomial in is a polynomial which is simultaneously a monomial in both and , when considered as a polynomial in with coefficients in (or as a polynomial in with coefficients in ). The degree of a monomial in is the sum of its individual degrees in the respective indeterminates and (in and ), and the degree of a polynomial in is the supremum of the degrees of its monomial summands, if it has any.
In three variables, we have , and in any finite number of variables, we have inductively , with monomials and degrees defined in analogy to the two variable case. In any number of variables, a polynomial ring is a graded ring with graded component equal to the -module generated by the monomials of degree .
3 Polynomial rings in arbitrarily many variables
For any nonempty set , let denote the set of all finite subsets of . For each element of , set . Any two elements satisfying give rise to the relationship if we consider to be embedded in in the obvious way. The union of the rings (or, more formally, the categorical direct limit of the direct system of rings ) is defined to be the ring .
|Date of creation||2013-03-22 11:52:27|
|Last modified on||2013-03-22 11:52:27|
|Last modified by||djao (24)|