resultant (alternative treatment)
Let be a field and let
be two polynomials over of degree and , respectively. We define , the resultant of and , to be the determinant of a square matrix with columns 1 to formed by shifted sequences consisting of the coefficients of , and columns to formed by shifted sequences consisting of coefficients of , i.e.
The resultant of two polynomials is non-zero if and only if the polynomials are relatively prime.
Proof. Let be two arbitrary polynomials of degree and , respectively. The polynomials are relatively prime if and only if every polynomial — including the unit polynomial 1 — can be formed as a linear combination of and . Let
be polynomials of degree and , respectively. The coefficients of the linear combination are given by the following matrix–vector multiplication:
The following Proposition describes the resultant of two polynomials in terms of the polynomials’ roots. Indeed this property uniquely characterizes the resultant, as can be seen by carefully studying the appended proof.
Let be as above and let and be their respective roots in the algebraic closure of . Then,
Proof. The multilinearity property of determinants implies that
Thus, let be indeterminates and set
Now by Proposition 1, vanishes if we replace any of the by any of and hence divides .
Next, consider the main diagonal of the matrix whose determinant gives . The first entries of the diagonal are equal to , and the next entries are equal to . It follows that the expansion of contains a term of the form . However, the expansion of contains exactly the same term, and therefore . Q.E.D.
|Title||resultant (alternative treatment)|
|Date of creation||2013-03-22 12:32:52|
|Last modified on||2013-03-22 12:32:52|
|Last modified by||Mathprof (13753)|