Degree and Euclidean division of polynomials
For the definition and characterization of the polynomials over a field , see Polynomials over a field.
Definition of the degree of a polynomial
The degree of a polynomial P, written , is an element of . The symbol is taken to have the usual properties of order and of addition relative to natural numbers.
The family is a -vector space basis of . For any , there exists an unique , only finitely many of which are nonzero, such that . If all the are zero, that is, if is the zero element of , then is defined as . If not all the are zero, since only finitely many are nonzero, there is a greatest value of such that . Then is defined as this greatest value.
Elementary properties of the degree
The following are easily checked to hold for any polynomials and (even when one or both are zero):
if , then
The notion of the degree of a polynomial allows us to formulate the following property of Euclidean division:
For all polynomials and , with nonzero, there exists exactly one pair of polynomials such that with .
For the proof, we fix the value of nonzero polynomial , and recurse over the degree of .
More specifically, being given, we consider the following proposition dependent on :
iff for all polynomials such that , there exist polynomials , with , such that .
is trivially true (for means that is zero, and satisfies with ).
Let us suppose true for some , and consider a polynomial the degree of which is less or equal to the successor of , that is to if , or otherwise.
If the degree of is less or equal to , then applies and there exist the wanted and .
If the degree of is less than that of , then we can directly write with and .
Remains the case in which is the successor of .