discriminant

discriminant

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 discriminant of

X

2

+

b

⁢

X

+

c

is

b

2

-

4

⁢

c

; the

quadratic formula

states that the roots are

-

b

/

2

±

b

2

-

4

⁢

c

/

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

http://planetmath.org/node/2895

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

=

x

3

+

A

⁢

x

+

B

, then its modular discriminant is the discriminant of the

cubic polynomial

on the right-hand side. For more on both these facts, see

[

1

]

on number fields and

[

2

]

on elliptic curves.

Definition.

The discriminant of order

n

∈

ℕ

is the polynomial, denoted
here

1

1

The discriminant of a polynomial

p

is oftentimes
also denoted as “

disc

(

p

)

”

by

δ

(

n

)

=

δ

(

n

)

⁢

(

a

1

,

…

,

a

n

)

, characterized by the following

relation

:

δ

(

n

)

⁢

(

s

1

,

s

2

,

…

,

s

n

)

=

∏

i

=

1

n

∏

j

=

i

+

1

n

(

x

i

-

x

j

)

2

,

(1)

where

s

k

=

s

k

⁢

(

x

1

,

…

,

x

n

)

,

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

s

1

,

…

,

s

n

.

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

/

a

n

Proposition 2

.

Up to sign, the discriminant is given by the

determinant

of a

2

⁢

n

-

1

square matrix

with columns 1 to

n

-

1

formed by shifting the

sequence

1

,

a

1

,

…

,

a

n

,  and columns

n

to

2

⁢

n

-

1

formed by
shifting the sequence

n

,

(

n

-

1

)

⁢

a

1

,

…

,

a

n

-

1

,  i.e.

δ

(

n

)

=

|

1

0

…

0

n

0

…

0

0

a

1

1

…

0

(

n

-

1

)

⁢

a

1

n

…

0

0

⋮

⋮

⋱

⋮

⋮

⋮

⋱

⋮

⋮

a

n

-

2

a

n

-

3

…

1

2

⁢

a

n

-

2

3

⁢

a

n

-

3

…

n

0

a

n

-

1

a

n

-

2

…

a

1

a

n

-

1

2

⁢

a

n

-

2

…

(

n

-

1

)

⁢

a

1

n

a

n

a

n

-

1

…

a

2

0

a

n

-

1

…

(

n

-

2

)

⁢

a

2

(

n

-

1

)

⁢

a

1

⋮

⋮

⋱

⋮

⋮

⋮

⋱

⋮

⋮

0

0

…

a

n

-

1

0

0

…

a

n

-

1

2

⁢

a

n

-

2

0

0

…

a

n

0

0

…

0

a

n

-

1

|

(2)

Multiple root test.

Let

𝕂

be a field, let

x

denote an

indeterminate

, and let

p

=

x

n

+

a

1

⁢

x

n

-

1

+

…

+

a

n

-

1

⁢

x

+

a

n

,

a

i

∈

𝕂

be a

monic polynomial

over

𝕂

. We define

δ

⁢

[

p

]

, the discriminant of

p

, by setting

δ

⁢

[

p

]

=

δ

(

n

)

⁢

(

a

1

,

…

,

a

n

)

.

The discriminant of a non-monic polynomial is defined homogenizing the
above definition, i.e by setting

δ

⁢

[

a

⁢

p

]

=

a

2

⁢

n

-

2

⁢

δ

⁢

[

p

]

,

a

∈

𝕂

.

Proposition 3

.

The discriminant vanishes if and only if

p

has multiple roots in
its

splitting field

.

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 (

2

) 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

)

=

a

1

2

-

4

⁢

a

2

δ

(

3

)

=

18

⁢

a

1

⁢

a

2

⁢

a

3

+

a

1

2

⁢

a

2

2

-

4

⁢

a

2

3

-

4

⁢

a

1

3

⁢

a

3

-

27

⁢

a

3

2

δ

(

4

)

=

a

1

2

⁢

a

2

2

⁢

a

3

2

-

4

⁢

a

2

3

⁢

a

3

2

-

4

⁢

a

1

3

⁢

a

3

3

+

18

⁢

a

1

⁢

a

2

⁢

a

3

3

-

27

⁢

a

3

4

-

4

⁢

a

1

2

⁢

a

2

3

⁢

a

4

+

16

⁢

a

2

4

⁢

a

4

+

18

⁢

a

1

3

⁢

a

2

⁢

a

3

⁢

a

4

-

80

⁢

a

1

⁢

a

2

2

⁢

a

3

⁢

a

4

-

6

⁢

a

1

2

⁢

a

3

2

⁢

a

4

+

144

⁢

a

2

⁢

a

3

2

⁢

a

4

-

27

⁢

a

1

4

⁢

a

4

2

+

144

⁢

a

1

2

⁢

a

2

⁢

a

4

2

-

128

⁢

a

2

2

⁢

a

4

2

-

192

⁢

a

1

⁢

a

3

⁢

a

4

2

+

256

⁢

a

4

3

Here is the matrix used to calculate

δ

(

4

)

:

δ

(

4

)

=

|

1

0

0

4

0

0

0

a

1

1

0

3

⁢

a

1

4

0

0

a

2

a

1

1

2

⁢

a

2

3

⁢

a

1

4

0

a

3

a

2

a

1

a

3

2

⁢

a

2

3

⁢

a

1

4

a

4

a

3

a

2

0

a

3

2

⁢

a

2

3

⁢

a

1

0

a

4

a

3

0

0

a

3

2

⁢

a

2

0

0

a

4

0

0

0

a

3

|

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

(

http://planetmath.org/BibliographyForNumberTheory

) and the

bibliography for algebraic geometry

(

http://planetmath.org/BibliographyForAlgebraicGeometry

).

Title

discriminant

Canonical name

Discriminant

Date of creation

2013-03-22 12:31:12

Last modified on

2013-03-22 12:31:12

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

17

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 12E05

Synonym

polynomial discriminant

Related topic

Resolvent

Related topic

DiscriminantOfANumberField

Related topic

ModularDiscriminant

Related topic

JInvariant

Related topic

DiscriminantOfAlgebraicNumber

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