biquadratic field

biquadratic field

A

biquadratic field

(or

biquadratic number field

) is a

biquadratic extension

of

ℚ

. To discuss these fields more easily, the set

S

will be defined to be the set of all

squarefree

integers

not equal to

1

. Thus, any biquadratic field is of the form

ℚ

⁢

(

m

,

n

)

for distinct

elements

m

and

n

of

S

.

Let

k

=

m

⁢

n

(

gcd

⁡

(

m

,

n

)

)

2

. It can easily be verified that

k

∈

S

,

k

≠

m

, and

k

≠

n

. Since

k

=

m

⁢

n

gcd

⁡

(

m

,

n

)

∈

ℚ

⁢

(

m

,

n

)

, the three distinct quadratic

subfields

of

ℚ

⁢

(

m

,

n

)

are

ℚ

⁢

(

m

)

,

ℚ

⁢

(

n

)

, and

ℚ

⁢

(

k

)

. Note that

ℚ

⁢

(

k

)

=

ℚ

⁢

(

m

⁢

n

)

.

Of the three

cyclotomic fields

of degree (

http://planetmath.org/ExtensionField

) four over

ℚ

,

ℚ

⁢

(

ω

8

)

and

ℚ

⁢

(

ω

12

)

are biquadratic fields. The quadratic subfields of

ℚ

⁢

(

ω

8

)

are

ℚ

⁢

(

2

)

,

ℚ

⁢

(

-

1

)

, and

ℚ

⁢

(

-

2

)

; the quadratic subfields of

ℚ

⁢

(

ω

12

)

are

ℚ

⁢

(

3

)

,

ℚ

⁢

(

-

1

)

, and

ℚ

⁢

(

-

3

)

.

Note that the only

rational prime

p

for which

e

(

P

|

p

)

=

4

is possible in a biquadratic field is

p

=

2

. (The notation

e

(

P

|

p

)

refers to the

ramification index

of the

prime ideal

P

over

p

.) This occurs for biquadratic fields

ℚ

⁢

(

m

,

n

)

in which exactly two of

m

,

n

, and

k

are

equivalent

(

http://planetmath.org/

Congruences

) to

2

⁢

mod

⁡

4

and the other is to

3

⁢

mod

⁡

4

. For example, in

ℚ

⁢

(

ω

8

)

=

ℚ

⁢

(

2

,

-

1

)

, we have that

e

(

P

|

2

)

=

4

.

Certain biquadratic fields provide excellent counterexamples to statements that some people might think to be true. For example, the biquadratic field

K

=

ℚ

⁢

(

2

,

-

3

)

is useful for demonstrating that a

subring

of a

principal ideal domain

need not be a principal ideal domain. It can easily be verified that

𝒪

K

(the

ring of integers

of

K

) is a principal ideal domain, but

ℤ

⁢

[

-

6

]

, which is a subring of

𝒪

K

, is not a principal ideal domain. Also, biquadratic fields of the form

L

=

ℚ

⁢

(

m

,

n

)

with

m

and

n

distinct elements of

S

such that

m

≡

1

⁢

mod

⁡

3

and

n

≡

1

⁢

mod

⁡

3

are useful for demonstrating that rings of integers need not have

power bases

over

ℤ

(

http://planetmath.org/PowerBasisOverMathbbZ

). Note that

3

splits completely in both

ℚ

⁢

(

m

)

and

ℚ

⁢

(

n

)

and thus in

L

. Therefore,

3

⁢

𝒪

L

=

P

1

⁢

P

2

⁢

P

3

⁢

P

4

for distinct prime ideals (

http://planetmath.org/PrimeIdeal

)

P

1

,

P

2

,

P

3

, and

P

4

of

𝒪

L

. Now suppose

𝒪

L

=

ℤ

⁢

[

α

]

for some

α

∈

L

. Then

L

=

ℚ

⁢

(

α

)

, and the

minimal polynomial

f

for

α

over

ℚ

has degree

4

. This yields that

f

, considered as a

polynomial

over

𝔽

3

, is supposed to factor into four distinct

monic polynomials

of degree

1

, which is a

contradiction

.

Title

biquadratic field

Canonical name

BiquadraticField

Date of creation

2013-03-22 15:56:24

Last modified on

2013-03-22 15:56:24

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

16

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 11R16

Synonym

biquadratic number field

Related topic

BiquadraticExtension

Related topic

BiquadraticEquation2

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