Let a = a0 + a1X +
··· + anXn,
b = b0 + b1X + ···
+ bmXm 
R[X] be two
polynomials in X. To define their sum and product it is convenient
to assume m = n. This can always be achieved by
adding terms of the form 0Xk.
Definition
The sum of the polynomials a and b is the polynomial
The sum of the polynomials a and b is the polynomial
a + b =
k = 0m (ak + bk)Xk.
The product of the two polynomials a and b is the polynomial
a · b = c0 + c1X + ··· + c2mX2m.
where ck = a0bk + a1bk - 1 + ··· + akb0.
R[X] provided with this addition
and multiplication is called the
polynomial ring
R[X].
In polynomial rings we encounter a new arithmetical structure. We will discuss division with remainder, gcd and the like in these rings.