- There exist polynomials
such that
(24)

(25) |

Conversely, let be a solution. Assume that and are coprime. Then would imply . This is impossible since and . Hence and are not coprime and holds.

(26) |

(27) |

(28) |

- is an isomorphism.
- If
then the Bézout coefficients
computed by the Extended Euclidean Algorithm
form the unique solution in
of the equation
(29)

(30) |

(31) |

- If (and still nonzero polynomials) we define .
- If or then .
- If then .

- .
- .
- there do not exist any
such that
(32)

(34) |

(35) |

Then the claim follows by cleaning up the denominators.

If then and are coprime in . Then there exists with stated degree bounds such that holds in . Observe that

- The coefficients of are in fact the unique solution of a linear system whose matrix is the Sylvester matrix of and .
- These coefficients can be computed by the Cramer's rule.
- Hence each of these coefficients is the quotient of a determinant of a submatrix of by .

*
*

2008-01-07