list of all imaginary quadratic extensions whose ring of integers is a PID

list of all imaginary quadratic extensions whose ring of integers is a PID

Gauss conjectured that for any

Δ

<

0

,

Δ

≡

0

,

1

(

mod

4

)

, then

𝒞

Δ

=

1

precisely when

Δ

=

-

3

,

-

4

,

-

7

,

-

8

,

-

11

,

-

12

,

-

16

,

-

19

,

-

27

,

-

28

,

-

43

,

-

67

,

-

163

In fact, he believed that as

Δ

→

-

∞

,

Δ

≡

0

,

1

(

mod

4

)

, so does the number of classes of (primitive

positive

integral

binary quadratic) forms with

http://planetmath.org/node/IntegralBinaryQuadraticForms

discriminant

Δ

.

It is relatively easy to show that the only

Δ

≡

0

(

mod

4

)

with this

property

are the ones in this list; that proof is given in an addendum to this article.

However, proving the

remainder

of Gauss’ hypotheses, regarding the odd values in the list, proved significantly harder. In the first half of the

20

th

century, Siegel showed that there was at most one such value beyond what Gauss had found. Heegner, Stark, and Baker showed, about

30

years later, that there are in fact no more (

[

2

]

,

[

3

]

,

[

4

]

).

Thus given an

imaginary quadratic extension

K

, it follows that the

ring of integers

of

K

, denoted

𝒪

K

, is a PID if and only if the

class group

of

K

is trivial if and only if there is only one class of primitive

quadratic forms

of the appropriate

http://planetmath.org/node/DiscriminantOfANumberField

discriminant

d

K

if and only if

d

K

is in the set above. So in particular, there are a finite number of imaginary quadratic extensions of

ℚ

whose ring of integers is a PID (and hence a UFD).

The values of

Δ

above that correspond to

𝒪

K

for some

K

are:

Δ

=

d

K

K

𝒪

K

-

3

ℚ

⁢

(

-

3

)

ℤ

⁢

[

1

+

-

3

2

]

-

4

ℚ

⁢

(

-

1

)

ℤ

⁢

[

-

1

]

-

7

ℚ

⁢

(

-

7

)

ℤ

⁢

[

1

+

-

7

2

]

-

8

ℚ

⁢

(

-

2

)

ℤ

⁢

[

-

2

]

-

11

ℚ

⁢

(

-

11

)

ℤ

⁢

[

1

+

-

11

2

]

-

19

ℚ

⁢

(

-

19

)

ℤ

⁢

[

1

+

-

19

2

]

-

43

ℚ

⁢

(

-

43

)

ℤ

⁢

[

1

+

-

43

2

]

-

67

ℚ

⁢

(

-

67

)

ℤ

⁢

[

1

+

-

67

2

]

-

163

ℚ

⁢

(

-

163

)

ℤ

⁢

[

1

+

-

163

2

]

We therefore get

Theorem 1

.

(Stark-Heegner)

If

d

<

0

, then the

class number

of

Q

⁢

(

d

)

is equal to

1

if and only if

d

=

-

1

,

-

2

,

-

3

,

-

7

,

-

11

,

-

19

,

-

43

,

-

67

,

or

-

163

(where

d

=

-

1

,

-

2

correspond to

Δ

=

-

4

,

-

8

and otherwise

d

=

Δ

).

How about the other four values

Δ

=

-

12

,

-

16

,

-

27

,

-

28

? Each of these corresponds to a non-maximal

http://planetmath.org/node/OrderInAnAlgebra

order in a

quadratic extension

(i.e. a proper subring of the

ring of algebraic integers

). Specifically, we have

Δ

=

-

12

↔

ℤ

⁢

[

2

⁢

1

+

-

3

2

]

=

ℤ

⁢

[

-

3

]

⊊

𝒪

K

⁢

for

⁢

K

=

ℚ

⁢

(

-

3

)

Δ

=

-

16

↔

ℤ

⁢

[

2

⁢

-

1

]

⊊

𝒪

K

⁢

for

⁢

K

=

ℚ

⁢

(

-

1

)

Δ

=

-

27

↔

ℤ

⁢

[

3

⁢

1

+

-

3

2

]

⊊

𝒪

K

⁢

for

⁢

K

=

ℚ

⁢

(

-

3

)

Δ

=

-

28

↔

ℤ

⁢

[

2

⁢

1

+

-

7

2

]

=

ℤ

⁢

[

-

7

]

⊊

𝒪

K

⁢

for

⁢

K

=

ℚ

⁢

(

-

7

)

Note that this does

not

mean that these rings are PIDs, since the

invertible ideals

in an order that is not the entire ring of integers do not include all ideals.

References

1

Cox, D.A.

Primes of the Form

x

2

+

n

⁢

y

2

: Fermat,

Class Field Theory

, and

Complex Multiplication

, Wiley 1997.

2

Heegner, K.,

Diophantische Analysis und Modulfunktionen

, Math. Zeit., 56 (1952), pp. 227-253.

3

Stark, H.M.,

A

complete

determination of the complex quadratic fields with class number one

, Mich. Math. J., 14 (1967), pp. 1-27.

4

Baker, A.,

Linear forms

in the logarithms of

algebraic numbers

, Mathematika, 13 (1966), pp. 204-216.

Title

list of all imaginary quadratic extensions whose ring of integers is a PID

Canonical

name

ListOfAllImaginaryQuadraticExtensionsWhoseRingOfIntegersIsAPID

Date of creation

2013-03-22 16:56:42

Last modified on

2013-03-22 16:56:42

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

7

Author

rm50 (10146)

Entry type

Result

Classification

msc 11E12

Classification

msc 11R29

Classification

msc 11E16

Related topic

EuclideanNumberField

Related topic

ImaginaryQuadraticField

Related topic

LemmaForImaginaryQuadraticFields

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