Polynomial ring/Field/Lemma of Bezout/Section

Theorem

Let ${}K$ be a field and let ${}P_{1},\ldots ,P_{n}$ denote polynomials over ${}K$. Let ${}G$ be a greatest common divisor of the ${}P_{i}$. Then there exists a representation

${}G=Q_{1}P_{1}+\cdots +Q_{n}P_{n}\,$

with

${}Q_{1},\ldots ,Q_{n}\in K[X]$.

Proof

We consider the set of all linear combinations

${}I={\left\{Q_{1}P_{1}+\cdots +Q_{n}P_{n}\mid Q_{i}\in K[X]\right\}}\,.$

This is an ideal of ${}K[X]$, as can be checked directly. Due to fact, this ideal is a principal ideal, hence,

${}I=(E)\,$

with a certain polynomial ${}E$. This ${}E$ is a common divisor of the ${}P_{i}$. Because of ${}P_{i}\in I=(E)$, we have

${}P_{i}=H_{i}E\,,$

that is ${}E$ is a factor of every ${}P_{i}$. A similar reasoning shows

${}P_{i}\in (G)\,$

for all ${}i$, and, therefore, also

${}(E)=I\subseteq (G)\,.$

Hence,

${}E=GF\,.$

By the condition, ${}G$ has the maximal degree among all common divisors. Therefore, ${}F\neq 0$ is a constant. Thus, we have

${}(G)=(E)=I\,,$

and, in particular, ${}G\in I$. Therefore, ${}G$ is a linear combination of the ${}P_{i}$.

$\Box$

Corollary

Let ${}K$ be a field and let ${}P_{1},\ldots ,P_{n}$ denote coprime polynomials over ${}K$. Then there exists a representation

${}Q_{1}P_{1}+\cdots +Q_{n}P_{n}=1\,$

with

${}Q_{1},\ldots ,Q_{n}\in K[X]$.

Proof

This follows directly from fact.

$\Box$

Remark

For given polynomials ${}P_{1},\ldots ,P_{n}\in K[X]$, we can determine explicitly their greatest common divisor ${}G$ and a representation

${}G=Q_{1}P_{1}+\cdots +Q_{n}P_{n}\,$

as stated in fact. For this, we restrict ourselves to the case ${}n=2$. Let the degree of ${}P_{1}$ be as large as the degree of ${}P_{2}$. The division with remainder yields

${}P_{1}=P_{2}H_{2}+R_{2}\,$

with a remaining polynomial, whose degree is smaller than the degree of ${}P_{2}$, or which is ${}0$. The main point is that the ideals

${}(P_{1},P_{2})=(P_{2},R_{2})\,$

are identical and, thus, the greatest common divisor of ${}P_{1}$ and ${}P_{2}$ and of ${}P_{2}$ and ${}R_{2}$ coincide. Now we perform again the division with remainder, dividing ${}P_{2}$ by ${}R_{2}$ with remainder ${}R_{3}$, and again the ideal ${}(R_{2},R_{3})$ coincides with the starting ideal. In this way, we obtain a sequence of remaining polynomials

$R_{0}=P_{1},R_{1}=P_{2},R_{2},\ldots ,R_{k}\neq 0,0,$

with the property that two adjacent polynomials generate the same ideal. Hence, ${}R_{k}$ (the last remaining polynomial different from ${}0$) is the greatest common divisor of ${}R_{0}$ and ${}R_{1}$. We can find a representation of ${}R_{k}$ as a linear combination of the ${}R_{0},R_{1}$, by working back this algorithm using the equations which describe the divisions with remainder.

The method described in the previous remark is called Euclidean algorithm. It holds accordingly also for the integers.

Example

We want to determine the greatest common divisor for the polynomials ${}X^{2}+X+1$ and ${}X-1$ in ${}\mathbb {Q} [X]$. For this, we perform the division with remainder, and we obtain

${}X^{2}+X+1={\left(X+2\right)}{\left(X-1\right)}+3\,.$

Thus the two polynomials are coprime. A representation of the ${}1$ is

${}1={\frac {1}{3}}{\left(X^{2}+X+1\right)}-{\frac {1}{3}}{\left(X+2\right)}{\left(X-1\right)}\,.$