SECTION 3.7 Discriminant 177
S(f,g) =
an an− 1 an− 2 ··· 0 0 0
0 an an− 1 ··· 0 0 0
... ... ... ... ... ...
0 0 0 ··· a 1 a 0 0
0 0 0 ··· a 2 a 1 a 0
bm bm− 1 bm− 2 ··· 0 0 0
0 bm bm− 1 ··· 0 0 0
... ... ... ... ... ...
0 0 0 ··· b 1 b 0 0
0 0 0 ··· b 2 b 1 b 0
.
The resultantR(f,g) of f(x) andg(x) is the determinant of the
corresponding Sylvester matrix:
R(f,g) = detS(f,g).
For example, iff(x) =a 2 x^2 +a 1 x+a 0 andg(x) =b 3 x^3 +b 2 x^2 +b 1 x+b 0 ,
then
S(f,g) = det
a 2 a 1 a 0 0 0
0 a 2 a 1 a 0 0
0 0 a 2 a 1 a 0
b 3 b 2 b 1 b 0 0
0 b 3 b 2 b 1 b 0
.
Note that the resultant of two polynomials clearly remains unchanged
upon field extension.
We aim to list a few simple—albeit technical—results about the
resultant. The first is reasonably straightforward and is proved by just
keeping track of the sign changes introduced by swapping rows in a
determinant.
Lemma 2.
R(f,g) = (−1)mnR(g,f)
wheredegf=nanddegg=m.