calculating the splitting of primes

calculating the splitting of primes

Let

K

|

L

be an

extension

of

number fields

, with

rings of integers

𝒪

K

,

𝒪

L

. Since this extension is

separable

(

http://planetmath.org/SeparablePolynomial

), there exists

α

∈

K

with

L

⁢

(

α

)

=

K

and by multiplying by a suitable

integer

, we may assume that

α

∈

𝒪

K

(we do

not

require that

𝒪

L

⁢

[

α

]

=

𝒪

K

. There is not, in general, an

α

∈

𝒪

L

with this property). Let

f

∈

𝒪

L

⁢

[

x

]

be the

minimal polynomial

of

α

.

Now, let

𝔭

be a

prime ideal

of

L

that does not divide

Δ

⁢

(

f

)

⁢

Δ

⁢

(

𝒪

K

)

-

1

, and let

f

¯

∈

𝒪

L

/

𝔭

⁢

𝒪

L

⁢

[

x

]

be the

reduction

of

f

mod

𝔭

, and let

f

¯

=

f

¯

1

⁢

⋯

⁢

f

¯

n

be its

factorization

into

irreducible polynomials

. If there are repeated factors, then

p

splits in

K

as the

product

𝔭

=

(

𝔭

,

f

1

⁢

(

α

)

)

⁢

⋯

⁢

(

𝔭

,

f

n

⁢

(

α

)

)

,

where

f

i

is any

polynomial

in

𝒪

L

⁢

[

x

]

reducing to

f

¯

i

. Note that in this case

𝔭

is

unramified

, since all

f

i

are

pairwise coprime

mod

𝔭

For example, let

L

=

ℚ

,

K

=

ℚ

⁢

(

d

)

where

d

is a square-free integer.
Then

f

=

x

2

-

d

. For any prime

𝔭

,

f

is

irreducible

mod

𝔭

if and only if it has no roots mod

𝔭

, i.e.

d

is a

quadratic non-residue

mod

𝔭

. Using

quadratic reciprocity

, we can obtain a

congruence

condition mod

4

⁢

p

for which primes split and which do not. In general, this is possible for all fields with

abelian

Galois groups

, using field .

Furthermore, let

K

′

be the

splitting field

of

L

. Then

G

=

Gal

(

K

′

|

L

)

acts on the roots of

f

, giving a map

G

→

S

m

, where

m

=

deg

⁡

f

. Given a prime

𝔭

of

𝒪

L

, the

Artin symbol

[

𝔓

,

K

′

|

L

]

for any

𝔓

lying over

𝔭

is determined up to conjugacy by

𝔭

. Its in

S

n

is a product of disjoint cycles of length

m

1

,

…

,

m

n

where

m

i

=

deg

⁡

f

i

.
This is useful not just for prime splitting, but also for the calculation of Galois groups.

Another useful fact is the

Frobenius theorem

, which that every element of

G

is

[

𝔓

,

K

′

|

L

]

for infinitely many primes

𝔓

of

𝒪

K

′

.

For example, let

f

=

x

3

+

x

2

+

2

∈

ℤ

⁢

[

x

]

. This is irreducible mod 3, and thus irreducible.

Galois theory

tells us that

G

=

Gal

(

K

′

|

L

)

is a

subgroup

of

S

3

, and so is

isomorphic

to

C

3

or

S

3

, but it is not obvious which. But if we consider

p

=

7

,

f

≡

(

x

-

2

)

⁢

(

x

2

+

3

⁢

x

-

1

)

(

mod

7

)

, and the quadratic factor is irreducible mod 7. Thus,

G

≅

S

3

.

Or let

f

=

x

4

+

a

⁢

x

2

+

b

for some integers

a

,

b

and is irreducible. For a prime

p

, consider the factorization of

f

. Either it remains irreducible (

G

contains a 4-cycle), splits as the product of irreducible quadratics (

G

contains a cycle of the form

(

12

)

⁢

(

34

)

) or

f

¯

has a root. If

β

is a root of

f

, then so is

-

β

, and so assuming

p

≠

2

, there are at least two roots, and so a 3-cycle is impossible. Thus

G

≅

C

4

or

D

4

.

Title

calculating the splitting of primes

Canonical name

CalculatingTheSplittingOfPrimes

Date of creation

2013-03-22 13:53:24

Last modified on

2013-03-22 13:53:24

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

12

Author

mathcam (2727)

Entry type

Topic

Classification

msc 11R04

Related topic

PrimeIdealDecompositionInQuadraticExtensionsOfMathbbQ

Related topic

PrimeIdealDecompositionInCyclotomicExtensionsOfMathbbQ

Related topic

NumberField

Related topic

SplittingAndRamificationInNumberFieldsAndGaloisExtensions

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