- ,
- ,
- ,
- .

- ,
- ,
- for every we have .

First we prove that must have degree 0 w.r.t. . This is obvious if . So let us assume that holds. This implies . If would have positive degree w.r.t. , then, by the properties of a regular Euclidean size, we would have . Since holds (by definition of an Euclidean domain) we would have a contradiction with . Therefore in any case we have .

Next, let us prove the proposition for those elements which have quotient 0 w.r.t. . Observe that if one quotient of w.r.t. is zero then we have . Hence every quotient of w.r.t. is null. (Otherwise we would have by the properties of a regular Euclidean size.) Since and since has degree 0 w.r.t. , the element has degree 0 w.r.t. and is a -adic expansion of w.r.t. . Therefore the proposition is proved in the case of the elements which have quotient 0 w.r.t. .

We can assume now that does not have quotient 0 w.r.t. . Then, two cases arise:

- Either both and hold, and this case we have ; then, we consider a couple quotient-remainder given as in Definition 2 when dividing w.r.t. .
- Or
. Then, there exists a smallest integer
such that the remainder in an division of
w.r.t.
is not zero. (This follows from the fact that
is a
Unique Factorization Domain and
is a prime.)
Let
be a couple quotient-remainder
in an division of
w.r.t.
with
.
- If , then we stop.
- If , then ; then, we consider a couple quotient-remainder given as in Definition 2 when dividing w.r.t. .

Continuing in this manner we obtain a finite sequence , a finite sequence and a finite sequence such that we have

This process stops since we have

(1) |

and since the Euclidean size of an element has value in . Clearly, there exitst such that writes

where are null or have degree 0 w.r.t. , by the first part of this proof. Moreover holds.

*As we shall see now, it extends to a ring
of univariate polynomials over a commutative ring
with identity element and to the element
where
is an element of
.
*

(2) |

Since the conclusion (existence and unicity) follows by the induction hypothesis.

- ,
- ,
- .

where . The division of by shows that . Similarly, from (Equation 3), we have . Plugging and in (Equation 5) leads to

for a polynomial .

- First, in Proposition 4, one can consider as a new indeterminate, and then the result holds for a polynomial .
- Secondly, the notion of a formal derivative of a univariate polynomial (over a commutative ring with identity element) leads naturally to the notion of a partial derivative of a multivariate polynomial w.r.t. a variable. We will not detail this here, since this is quite straightforward
- The quantity in Relation (4) lies in the ideal .

(7) |

(8) |

(9) |

(10) |

(11) |

*
*

2008-01-07