examples of integrally closed extensions

examples of integrally closed extensions

Example.

ℤ

⁢

[

5

]

is not

integrally closed

, for

u

=

1

+

5

2

∈

ℚ

⁢

[

5

]

is integral over

ℤ

⁢

[

5

]

since

u

2

-

u

-

1

=

0

, but

u

∉

ℤ

⁢

[

5

]

.

Example.

R

=

ℤ

⁢

[

2

,

3

]

is not integrally closed. Note that

(

6

+

2

)

/

2

∉

R

, but that

(

6

+

2

2

)

2

=

2

+

3

and so

(

6

+

2

)

/

2

is integral over

ℤ

since it satisfies the

polynomial

(

z

2

-

2

)

2

-

3

=

0

.

Example.

𝒪

K

is integrally closed when

[

K

:

ℚ

]

<

∞

. For if

u

∈

K

is integral over

𝒪

K

, then

ℤ

⊂

𝒪

K

⊂

𝒪

K

⁢

[

u

]

are all

integral extensions

, so

u

is integral over

ℤ

, so

u

∈

𝒪

K

by definition. In fact,

𝒪

K

can be defined as the

integral closure

of

ℤ

in

K

.

Example.

ℂ

⁢

[

x

,

y

]

/

(

y

2

-

x

3

)

. This is a domain because

y

2

-

x

3

is

irreducible

hence a

prime ideal

. But this

quotient ring

is not integrally closed. To see this, parameterize

ℂ

⁢

[

x

,

y

]

→

ℂ

⁢

[

t

]

by

x

↦

t

2

y

↦

t

3

The kernel of this map is

(

y

2

-

x

3

)

, and its image is

ℂ

⁢

[

t

2

,

t

3

]

. Hence

ℂ

⁢

[

x

,

y

]

/

(

y

2

-

x

3

)

≅

ℂ

⁢

[

t

2

,

t

3

]

and the

field of fractions

of the latter ring is obviously

ℂ

⁢

(

t

)

. Now,

t

is integral over

ℂ

⁢

[

t

2

,

t

3

]

(

z

2

-

t

2

is its polynomial), but is not in

ℂ

⁢

[

t

2

,

t

3

]

.

t

corresponds to

y

x

in the original ring

ℂ

⁢

[

x

,

y

]

/

(

y

2

-

x

3

)

, which is thus not integrally closed (the

minimal polynomial

of

y

x

is

z

2

-

x

since

(

y

x

)

2

-

x

=

y

2

x

2

-

x

=

x

3

x

2

-

x

=

0

). The failure of integral closure in this

coordinate ring

is due to a codimension 1 singularity of

y

2

-

x

3

at

0

.

Example.

A

=

ℂ

⁢

[

x

,

y

,

z

]

/

(

z

2

-

x

⁢

y

)

is integrally closed. For again, parameterize

A

→

ℂ

⁢

[

u

,

v

]

by

x

↦

u

2

y

↦

v

2

z

↦

u

⁢

v

The kernel of this map is

z

2

-

x

⁢

y

and its image is

B

=

ℂ

⁢

[

u

2

,

v

2

,

u

⁢

v

]

. Claim

B

is integrally closed. We prove this by showing that the integral closure of

ℂ

⁢

[

x

,

y

]

in

ℂ

⁢

(

x

,

y

,

x

⁢

y

)

is

ℂ

⁢

[

x

,

y

,

x

⁢

y

]

. Choose

r

+

s

⁢

x

⁢

y

∈

ℂ

⁢

(

x

,

y

,

x

⁢

y

)

,

r

,

s

∈

ℂ

⁢

(

x

,

y

)

such that

r

+

s

⁢

x

⁢

y

is integral over

ℂ

⁢

[

x

,

y

]

. Then

r

-

s

⁢

x

⁢

y

is also integral over

ℂ

⁢

[

x

,

y

]

, so their sum is. Hence

2

⁢

r

is integral over

ℂ

⁢

[

x

,

y

]

. But

ℂ

⁢

[

x

,

y

]

is a UFD, hence integrally closed, so

2

⁢

r

∈

ℂ

⁢

[

x

,

y

]

and thus

r

∈

ℂ

⁢

[

x

,

y

]

. Similarly,

s

⁢

x

⁢

y

is integral over

ℂ

⁢

[

x

,

y

]

, hence

s

2

⁢

x

⁢

y

∈

ℂ

⁢

[

x

,

y

]

,

s

∈

ℂ

⁢

(

x

,

y

)

. Clearly, then,

s

can have no

denominator

, so

s

∈

ℂ

⁢

[

x

,

y

]

. Hence

r

+

s

⁢

x

⁢

y

∈

ℂ

⁢

[

x

,

y

,

x

⁢

y

]

.

Title

examples of integrally closed extensions

Canonical name

ExamplesOfIntegrallyClosedExtensions

Date of creation

2013-03-22 17:01:32

Last modified on

2013-03-22 17:01:32

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

9

Author

rm50 (10146)

Entry type

Example

Classification

msc 13B22

Classification

msc 11R04

Generated on Fri Feb 9 17:25:50 2018 by

LaTeXML