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
Euclidean division
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
.