proof of casus irreducibilis for real fields

proof of casus irreducibilis for real fields

The classical statement of the

casus irreducibilis

is that if

f

⁢

(

x

)

is an

irreducible

cubic polynomial

with

rational

coefficients

and three

real roots

, then the roots of

f

⁢

(

x

)

are not

expressible

using real

radicals

. One example of such a

polynomial

is

x

3

-

3

⁢

x

+

1

, whose roots are

2

⁢

cos

⁡

(

2

⁢

π

/

9

)

,

2

⁢

cos

⁡

(

8

⁢

π

/

9

)

,

2

⁢

cos

⁡

(

14

⁢

π

/

9

)

.

This article generalizes the classical case to include all polynomials whose degree is not a power of

2

, and also generalizes the

base field

to be any real

extension

of

ℚ

:

Theorem 1

.

Let

F

⊂

R

be a field, and assume

f

⁢

(

x

)

∈

F

⁢

[

x

]

is an

irreducible polynomial

whose

splitting field

L

is real with

F

⊂

L

⊂

R

. Then the following are

equivalent

:

1.

Some root of

f

⁢

(

x

)

is expressible by real radicals over

F

;

2.

All roots of

f

⁢

(

x

)

are expressible by real radicals over

F

using only

square roots

;

3.

F

⊂

L

is a

radical extension

;

4.

[

L

:

F

]

is a power of

2

.

Proof.

That

(

2

)

⇒

(

1

)

is obvious, and

(

3

)

⇒

(

1

)

since

F

⊂

L

is radical, and is real since

L

⊂

ℝ

.

(

4

)

implies that

G

=

Gal

⁡

(

L

/

F

)

has order a power of

2

. Since

G

is a

2

-group, it has a nontrivial center (this follows directly from the

class equation

, or look here (

http://planetmath.org/ANontrivialNormalSubgroupOfAFinitePGroupGAndTheCenterOfGHaveNontrivialIntersection

)) and thus has a

normal subgroup

H

of order

2

, which corresponds to a

subfield

M

of

L

Galois over

F

with

[

L

:

M

]

=

2

. But then

Gal

⁡

(

M

/

F

)

is also a

2

-group, so inductively we see that we can write

F

=

K

0

⊂

K

1

⊂

…

⊂

K

m

-

1

=

M

⊂

K

m

=

L

where

[

K

i

:

K

i

-

1

]

=

2

. Thus each

K

i

is obtained from

K

i

-

1

by adjoining a square root; it must be a real square root since

L

⊂

ℝ

. This shows that

(

4

)

⇒

(

2

)

and

(

3

)

.

The meat of the proof is in showing that

(

1

)

⇒

(

4

)

. Let the roots of

f

⁢

(

x

)

be

α

1

,

…

,

α

m

, and assume, by renumbering if necessary, that

α

=

α

1

lies in a real radical extension

K

of

F

but that

[

L

:

F

]

is not a power of

2

. Choose an

odd prime

p

dividing

[

L

:

F

]

=

|

G

|

, and choose an

element

τ

∈

G

of order

p

. Then

τ

is not the

identity

, so for some

i

,

τ

⁢

(

α

i

)

≠

α

i

. Also, since

f

⁢

(

x

)

is irreducible,

G

acts transitively on the roots of

f

⁢

(

x

)

, so for some

ν

∈

G

,

ν

⁢

(

α

)

=

α

i

. Then

σ

=

ν

-

1

⁢

τ

⁢

ν

does not fix

α

, since

ν

-

1

⁢

τ

⁢

ν

⁢

(

α

)

=

ν

-

1

⁢

τ

⁢

(

α

i

)

≠

ν

-

1

⁢

(

α

i

)

=

α

Let

N

=

L

σ

be the

fixed field

of

σ

. Then

L

is Galois over

N

, and clearly

[

L

:

N

]

=

p

. But

Galois subfields of real radical extensions are at most quadratic

, so

L

cannot lie in a real radical extension of

N

.

However,

α

∉

N

,

α

∈

L

, and

[

L

:

N

]

is prime. Thus

L

=

N

⁢

(

α

)

⊂

N

⁢

K

(since

α

∈

K

). Additionally, since

F

⊂

F

⁢

(

α

)

⊂

K

is a real radical extension of

F

, we have also that

N

⁢

K

is a real radical extension of

N

⁢

F

=

N

. So

L

lies in the real radical extension

N

⁢

K

of

N

. But this is a

contradiction

and thus

[

L

:

F

]

must be a power of

2

.

One

consequence

of this

theorem

is the fact that if

f

⁢

(

x

)

∈

F

⁢

[

x

]

has degree not a power of

2

, then if

f

⁢

(

x

)

has all real roots, those roots are not expressible in terms of real radicals. If

deg

⁡

f

=

3

, we recover the original

casus irreducibilis

.

References

1

D.A. Cox,

Galois Theory

, Wiley-Interscience, 2004.

Title

proof of casus irreducibilis for real fields

Canonical name

ProofOfCasusIrreducibilisForRealFields

Date of creation

2013-03-22 17:43:08

Last modified on

2013-03-22 17:43:08

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

8

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 12F10

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