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discriminant∗
rspuzio†
2013-03-21 13:50:45
Summary. The discriminant of a given polynomial is a number, calculated
from the coefficients of that polynomial, that vanishes if and only if that polynomial has one or more multiple roots. Using the discriminant we can test for
the presence of multiple roots, without having to actually calculate the roots of
the polynomial in question.
There are other ways to do this of course; one can look at the formal derivative of the polynomial (it will be coprime to the original polynomial if and
only if that original had no multiple roots). But the discriminant turns out to
be valuable in a number of other contexts. For example, we will see that the
2
discriminant of X√
+ bX + c is b2 − 4c; the quadratic formula states that the
roots are −b/2 ± b2 − 4c/2, so that the discriminant also determines whether
the roots of this polynomial are real or not. In higher degrees, its role is more
complicated.
There are other uses of the word “discriminant” that are closely related
to this one. If Q(α) is a number field, then the discriminant of Q(α) is the
discriminant of the minimal polynomial of α. For more general extensions of
number fields, one must use a different definition of discriminant generalizing
this one. If we have an elliptic curve over the rational numbers defined by the
equation y 2 = x3 + Ax + B, then its modular discriminant is the discriminant
of the cubic polynomial on the right-hand side. For more on both these facts,
see [?] on number fields and [?] on elliptic curves.
Definition. The discriminant of order n ∈ N is the polynomial, denoted here
1
by δ (n) = δ (n) (a1 , . . . , an ), characterized by the following relation:
δ (n) (s1 , s2 , . . . , sn ) =
n
n
Y
Y
(xi − xj )2 ,
(1)
i=1 j=i+1
∗ hDiscriminanti
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1 The discriminant of a polynomial p is oftentimes also denoted as “disc(p)”
1
where
sk = sk (x1 , . . . , xn ),
k = 1, . . . , n
is the k th elementary symmetric polynomial.
The above relation is a defining one, because the right-hand side of (1)
is, evidently, a symmetric polynomial, and because the algebra of symmetric
polynomials is freely generated by the basic symmetric polynomials, i.e. every
symmetric polynomial arises in a unique fashion as a polynomial of s1 , . . . , sn .
Proposition 1. The discriminant d of a polynomial may be expressed with the
resultant R of the polynomial and its first derivative:
d = (−1)
n(n−1)
2
R/an
Proposition 2. Up to sign, the discriminant is given by the determinant of
a 2n−1 square matrix with columns 1 to n−1 formed by shifting the sequence
1, a1 , . . . , an , and columns n to 2n−1 formed by shifting the sequence n, (n−
1)a1 , . . . , an−1 , i.e.
δ (n)
1
a1
.
.
.
an−2
= an−1
an
..
.
0
0
0
1
..
.
an−3
an−2
an−1
..
.
0
0
Multiple root test.
...
...
..
.
...
...
...
..
.
...
...
0
0
..
.
1
a1
a2
..
.
an−1
an
n
(n − 1) a1
..
.
2 an−2
an−1
0
..
.
0
0
0
n
..
.
3 an−3
2 an−2
an−1
..
.
0
0
...
...
..
.
...
...
...
..
.
...
...
0
0
..
.
n
(n − 1)a1
(n − 2)a2
..
.
an−1
0
0
0
..
.
0
n
(n − 1)a1 ..
.
2 an−2 an−1
(2)
Let K be a field, let x denote an indeterminate, and let
p = xn + a1 xn−1 + . . . + an−1 x + an ,
ai ∈ K
be a monic polynomial over K. We define δ[p], the discriminant of p, by setting
δ[p] = δ (n) (a1 , . . . , an ) .
The discriminant of a non-monic polynomial is defined homogenizing the above
definition, i.e by setting
δ[ap] = a2n−2 δ[p],
a ∈ K.
Proposition 3. The discriminant vanishes if and only if p has multiple roots
in its splitting field.
2
Proof. It isn’t hard to show that a polynomial has multiple roots if and only if
that polynomial and its derivative share a common root. The desired conclusion
now follows by observing that the determinant formula in equation (??) gives
the resolvent of a polynomial and its derivative. This resolvent vanishes if and
only if the polynomial in question has a multiple root.
Some Examples.
Here are the first few discriminants.
δ (1) = 1
δ (2) = a21 − 4 a2
δ (3) = 18 a1 a2 a3 + a21 a22 − 4 a32 − 4 a31 a3 − 27a23
δ (4) = a21 a22 a23 − 4 a32 a23 − 4 a31 a33 + 18 a1 a2 a33 − 27 a43
− 4 a21 a32 a4 + 16 a42 a4 + 18 a31 a2 a3 a4 − 80 a1 a22 a3 a4
− 6 a21 a23 a4 + 144 a2 a23 a4 − 27 a41 a24 + 144 a21 a2 a24
− 128 a22 a24 − 192 a1 a3 a24 + 256 a34
Here is the matrix used to calculate δ (4) :
1 0 0
4
a1 1 0 3a1
a2 a1 1 2a2
(4)
δ = a3 a2 a1 a3
a4 a3 a2
0
0 a4 a3
0
0 0 a4
0
0
4
3a1
2a2
a3
0
0
0
0
4
3a1
2a2
a3
0
0
0
0
4
3a1
2a2
a3
References
[1] Daniel A. Marcus, Number Fields, Springer, New York.
[2] Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag,
New York, 1986.
See also the bibliography for number theory and the bibliography for algebraic geometry.
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