Euler’s derivation of the quartic formula

Euler’s derivation of the quartic formula

Let us consider the

quartic equation

y

4

+

p

⁢

y

2

+

q

⁢

y

+

r

=

0

,

(1)

where

p

,

q

,

r

are arbitrary known

complex numbers

.  We substitute in the equation

y

:=

u

+

v

+

w

.

(2)

We get firstly

y

2

=

(

u

2

+

v

2

+

w

2

)

+

2

⁢

(

v

⁢

w

+

w

⁢

u

+

u

⁢

v

)

,

y

4

=

(

u

2

+

v

2

+

w

2

)

2

+

4

⁢

(

u

2

+

v

2

+

w

2

)

⁢

(

v

⁢

w

+

w

⁢

u

+

u

⁢

v

)

+

4

⁢

(

v

2

⁢

w

2

+

w

2

⁢

u

2

+

u

2

⁢

v

2

)

+

8

⁢

u

⁢

v

⁢

w

⁢

(

u

+

v

+

w

)

.

Thus (1) attains the form

4

⁢

(

v

2

⁢

w

2

+

w

2

⁢

u

2

+

u

2

⁢

v

2

)

+

(

u

2

+

v

2

+

w

2

)

2

+

p

⁢

(

u

2

+

v

2

+

w

2

)

+

r

+

(

v

⁢

w

+

w

⁢

u

+

u

⁢

v

)

⁢

[

4

⁢

(

u

2

+

v

2

+

w

2

)

+

2

⁢

p

]

+

(

u

+

v

+

w

)

⁢

[

8

⁢

u

⁢

v

⁢

w

+

q

]

=

0

.

When

u

,

v

,

w

are determined so that

u

2

+

v

2

+

w

2

=

-

p

2

,

(3)

u

⁢

v

⁢

w

=

-

q

8

,

(4)

the expressions in the brackets vanish and our equation shrinks to the form

v

2

⁢

w

2

+

w

2

⁢

u

2

+

u

2

⁢

v

2

=

p

2

-

4

⁢

r

16

.

(5)

Squaring (4) gives

u

2

⁢

v

2

⁢

w

2

=

q

2

64

.

(6)

The left hand sides of (3), (5) and (6) are the

elementary symmetric polynomials

of

u

2

,

v

2

,

w

2

, whence these three squares are the roots

z

1

,

z

2

,

z

3

of the so-called

cubic resolvent

equation

z

3

+

p

2

⁢

z

2

+

p

2

-

4

⁢

r

16

⁢

z

-

q

2

64

=

0

.

(7)

Therefore we may write

u

=

±

z

1

,

v

=

±

z

2

,

w

=

±

z

3

.

All 8 sign

combinations

of those

square roots

satisfy

the equations (3), (5), (6). In order to satisfy also (4) the signs must be chosen suitably.  If

u

0

,

v

0

,

w

0

is some suitable combination of the values of the square roots, then all possible combinations are

u

0

,

v

0

,

w

0

;

u

0

,

-

v

0

,

-

w

0

;

-

u

0

,

v

0

,

-

w

0

;

-

u

0

,

-

v

0

,

w

0

.

Accordingly, we have the

Theorem

(Euler 1739).  The roots of the equation (1) are

{

y

1

=

u

0

+

v

0

+

w

0

,

y

2

=

u

0

-

v

0

-

w

0

,

y

3

=

-

u

0

+

v

0

-

w

0

,

y

4

=

-

u

0

-

v

0

+

w

0

,

(8)

where

u

0

,

v

0

,

w

0

are square roots of the roots of the cubic resolvent (7).  The signs of the square roots must be chosen such that

u

0

⁢

v

0

⁢

w

0

=

-

q

8

.

The equations (8) imply an important

formula

(

y

1

-

y

2

)

⁢

(

y

1

-

y

3

)

⁢

(

y

1

-

y

4

)

⁢

(

y

2

-

y

3

)

⁢

(

y

2

-

y

4

)

⁢

(

y

3

-

y

4

)

=

-

2

6

⁢

(

v

0

2

-

w

0

2

)

⁢

(

w

0

2

-

u

0

2

)

⁢

(

u

0

2

-

v

0

2

)

=

-

64

⁢

(

z

2

-

z

3

)

⁢

(

z

3

-

z

1

)

⁢

(

z

1

-

z

2

)

,

which yields the

Corollary.

A quartic equation has a

multiple root

always and only when its cubic resolvent has such one.

References

1

Ernst Lindelöf

:

Johdatus korkeampaan analyysiin

. Fourth edition. Werner Söderström Osakeyhtiö, Porvoo ja Helsinki (1956).

2

K. Väisälä

:

Lukuteorian ja korkeamman algebran alkeet

.  Tiedekirjasto No. 17. Kustannusosakeyhtiö Otava, Helsinki (1950).

Title

Euler’s

derivation

of the quartic formula

Canonical name

EulersDerivationOfTheQuarticFormula

Date of creation

2013-03-22 17:51:58

Last modified on

2013-03-22 17:51:58

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Theorem

Classification

msc 12D10

Synonym

quartic formula by Euler

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SymmetricQuarticEquation

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