aarjaneiro/planetmath huggingface dataset:
every PID is a UFD - alternative proof

every PID is a UFD - alternative proof

Proposition

.

If

R

is a

principal ideal domain

, then

R

is a

unique factorization domain

.

Proof.

Recall, that due to Kaplansky Theorem (see this article (

http://planetmath.org/EquivalentDefinitionsForUFD

) for details) it is enough to show that every nonzero

prime ideal

in

R

contains a

prime element

.

On the other hand, recall that an element

p

∈

R

is prime if and only if an ideal

(

p

)

generated by

p

is nonzero and prime.

Thus, if

P

is a nonzero prime ideal in

R

, then (since

R

is a PID) there exists

p

∈

R

such that

P

=

(

p

)

. This

completes

the proof.

□

Title

every PID is a UFD - alternative proof

Canonical name

EveryPIDIsAUFDAlternativeProof

Date of creation

2013-03-22 19:04:26

Last modified on

2013-03-22 19:04:26

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

5

Author

joking (16130)

Entry type

Theorem

Classification

msc 13F07

Classification

msc 16D25

Classification

msc 13G05

Classification

msc 11N80

Classification

msc 13A15

Generated on Fri Feb 9 17:30:57 2018 by

LaTeXML
number field that is not norm-Euclidean

number field that is not norm-Euclidean

Proposition.

The

real quadratic field

ℚ

⁢

(

14

)

is not

norm-Euclidean

.

Proof.

We take the number

γ

=

1

2

+

1

2

⁢

14

which is not

integer

of the field (

14

≡

2

(

mod

4

)

).  Antithesis:

γ

=

ϰ

+

δ

where

ϰ

=

a

+

b

⁢

14

is an integer of the field (

a

,

b

∈

ℤ

) and

|

N

⁢

(

δ

)

|

=

|

(

1

2

-

a

)

2

-

14

⁢

(

1

2

-

b

)

2

|

<

1

.

Thus we would have

|

(

2

⁢

a

-

1

)

2

-

14

⁢

(

2

⁢

b

-

1

)

2

⏟

E

|

<

4

.

And since

(

2

⁢

a

-

1

)

2

=

4

⁢

(

a

-

1

)

⁢

a

+

1

≡

1

(

mod

8

)

,  it follows

E

≡

1

-

14

⋅

1

≡

3

(

mod

8

)

,  i.e.

E

=

3

.  So we must have

(

2

⁢

a

-

1

)

2

≡

(

2

⁢

a

-

1

)

2

-

14

⁢

(

2

⁢

b

-

1

)

2

≡

3

(

mod

7

)

.

(1)

But

{

0

,

±

1

,

±

2

,

±

3

}

is a

complete residue system

modulo 7, giving  the set

{

1

,

2

,

4

}

of possible

quadratic residues

modulo 7.  Therefore (1) is impossible.  The antithesis is wrong, whence the theorem 1 of the parent entry (

http://planetmath.org/EuclideanNumberField

) says that the

number field

is not norm-Euclidean.

Note.

The function N used in the proof is the usual

N

:

r

+

s

14

↦

r

2

-

14

s

2

(

r

,

s

∈

ℚ

)

defined in the field

ℚ

⁢

(

14

)

.  The notion of norm-Euclidean number field is based on the norm (

http://planetmath.org/NormAndTraceOfAlgebraicNumber

).  There exists a fainter function, the so-called

Euclidean valuation

, which can be defined in the

maximal orders

of some

algebraic number fields

(

http://planetmath.org/NumberField

); such a maximal order, i.e. the

ring of integers

of the number field, is then a

Euclidean domain

.  The existence of a Euclidean valuation guarantees that the maximal order is a UFD and thus a PID.  Recently it has been shown the existence of the Euclidean domain

ℤ

⁢

[

1

+

69

2

]

in the field

ℚ

⁢

(

69

)

but the field is not norm-Euclidean.

The maximal order

ℤ

⁢

[

14

]

of

ℚ

⁢

(

14

)

has also been proven to be a Euclidean domain (Malcolm Harper 2004 in

Canadian Journal of Mathematics

).

Title

number field that is not norm-Euclidean

Canonical name

NumberFieldThatIsNotNormEuclidean

Date of creation

2013-03-22 16:56:56

Last modified on

2013-03-22 16:56:56

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

15

Author

pahio (2872)

Entry type

Example

Classification

msc 13F07

Classification

msc 11R21

Classification

msc 11R04

Related topic

UniqueFactorizationAndIdealsInRingOfIntegers

Generated on Fri Feb 9 17:31:04 2018 by

LaTeXML
proof of Bezout’s Theorem

proof of Bezout’s Theorem

Let

D

be an

integral domain

with an

Euclidean valuation

. Let

a

,

b

∈

D

not both 0. Let

(

a

,

b

)

=

{

a

⁢

x

+

b

⁢

y

|

x

,

y

∈

D

}

.

(

a

,

b

)

is an ideal in

D

≠

{

0

}

. We choose

d

∈

(

a

,

b

)

such that

μ

⁢

(

d

)

is the smallest

positive

value. Then

(

a

,

b

)

is generated by

d

and has the property

d

|

a

and

d

|

b

. Two

elements

x

and

y

in

D

are

associate

if and only if

μ

⁢

(

x

)

=

μ

⁢

(

y

)

. So

d

is unique up to a unit in

D

. Hence

d

is the

greatest common divisor

of

a

and

b

.

Title

proof of Bezout’s

Theorem

Canonical name

ProofOfBezoutsTheorem

Date of creation

2013-03-22 13:19:58

Last modified on

2013-03-22 13:19:58

Owner

Thomas Heye (1234)

Last modified by

Thomas Heye (1234)

Numerical id

7

Author

Thomas Heye (1234)

Entry type

Proof

Classification

msc 13F07

Generated on Fri Feb 9 17:31:12 2018 by

LaTeXML
proof that a Euclidean domain is a PID

proof that a Euclidean domain is a PID

Let

D

be a

Euclidean domain

, and let

𝔞

⊆

D

be a nonzero ideal. We show that

𝔞

is principal. Let

A

=

{

ν

⁢

(

x

)

:

x

∈

𝔞

,

x

≠

0

}

be the set of

Euclidean valuations

of the non-zero elements of

𝔞

. Since

A

is a non-empty set of non-negative integers, it has a minimum

m

. Choose

d

∈

𝔞

such that

ν

⁢

(

d

)

=

m

. Claim that

𝔞

=

(

d

)

. Clearly

(

d

)

⊆

𝔞

. To see the reverse inclusion, choose

x

∈

𝔞

. Since

D

is a Euclidean domain, there exist elements

y

,

r

∈

D

such that

x

=

y

⁢

d

+

r

with

ν

⁢

(

r

)

<

ν

⁢

(

d

)

or

r

=

0

. Since

r

∈

𝔞

and

ν

⁢

(

d

)

is minimal in

A

, we must have

r

=

0

. Thus

d

|

x

and

x

∈

(

d

)

.

Title

proof that a Euclidean domain is a PID

Canonical name

ProofThatAEuclideanDomainIsAPID

Date of creation

2013-03-22 12:43:11

Last modified on

2013-03-22 12:43:11

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

7

Author

rm50 (10146)

Entry type

Result

Classification

msc 13F07

Related topic

PID

Related topic

UFD

Related topic

IntegralDomain

Related topic

EuclideanValuation

Generated on Fri Feb 9 17:31:14 2018 by

LaTeXML
uniqueness of division algorithm in Euclidean domain

uniqueness of division algorithm in Euclidean domain

Theorem

.

Let

a

,

b

be non-zero elements of a

Euclidean domain

D

with the

Euclidean valuation

ν

.  The

incomplete quotient

q

and the remainder

r

of the division algorithm

a

=

q

⁢

b

+

r

where

r

=

0

⁢

or

⁢

ν

⁢

(

r

)

<

ν

⁢

(

b

)

are unique if and only if

ν

⁢

(

a

+

b

)

≦

max

⁡

{

ν

⁢

(

a

)

,

ν

⁢

(

b

)

}

.

(1)

Proof.

Assume first (1) for the

elements

a

,

b

of

D

.  If we had

{

a

=

q

⁢

b

+

r

with

r

=

0

∨

ν

⁢

(

r

)

<

ν

⁢

(

b

)

,

a

=

q

′

⁢

b

+

r

′

with

r

′

=

0

∨

ν

⁢

(

r

′

)

<

ν

⁢

(

b

)

and

r

′

≠

r

,

q

′

≠

q

,  then the properties of the Euclidean valuation (

http://planetmath.org/EuclideanValuation

) and the

assumption

yield the of

inequalities

ν

⁢

(

b

)

≦

ν

⁢

(

(

q

′

-

q

)

⁢

b

)

=

ν

⁢

(

r

′

-

r

)

≦

max

⁡

{

ν

⁢

(

r

′

)

,

ν

⁢

(

-

r

)

}

<

ν

⁢

(

b

)

which is impossible.  We must infer that

r

′

-

r

=

0

or

q

′

-

q

=

0

.  But these two conditions are

equivalent

(

http://planetmath.org/Equivalent3

).  Thus the division algorithm is unique.

Conversely

, assume that (1) is not true for non-zero elements

a

,

b

of

D

, i.e.

ν

⁢

(

a

+

b

)

>

max

⁡

{

ν

⁢

(

a

)

,

ν

⁢

(

b

)

}

.

Then we obtain two repsesentations

b

=

0

⁢

(

a

+

b

)

+

b

=

1

⁢

(

a

+

b

)

-

a

where

ν

⁢

(

b

)

<

ν

⁢

(

a

+

b

)

and

ν

⁢

(

-

a

)

=

ν

⁢

(

a

)

<

ν

⁢

(

a

+

b

)

.  Thus the incomplete quotient and the remainder are not unique.

Title

uniqueness of division algorithm in Euclidean domain

Canonical name

UniquenessOfDivisionAlgorithmInEuclideanDomain

Date of creation

2013-03-22 17:53:00

Last modified on

2013-03-22 17:53:00

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Theorem

Classification

msc 13F07

Related topic

KrullValuation

Related topic

Quotient

Defines

incomplete quotient

Generated on Fri Feb 9 17:31:23 2018 by

LaTeXML
example of Smith normal form

example of Smith normal form

As an example, we will find the

Smith normal form

of the following matrix over the integers.

(

2

4

4

-

6

6

12

10

-

4

-

16

)

The following matrices are the intermediate steps as the

algorithm

is applied to the above matrix.

(

2

0

0

-

6

18

24

10

-

24

-

36

)

→

(

2

0

0

0

18

24

0

-

24

-

36

)

→

(

2

0

0

0

18

24

0

-

6

-

12

)

→

(

2

0

0

0

6

12

0

18

24

)

→

(

2

0

0

0

6

12

0

0

-

12

)

→

(

2

0

0

0

6

0

0

0

12

)

So the Smith normal form is

(

2

0

0

0

6

0

0

0

12

)

and the elementary divisors are

2

,

6

and

12

.

Title

example of Smith normal form

Canonical name

ExampleOfSmithNormalForm

Date of creation

2013-03-22 14:22:51

Last modified on

2013-03-22 14:22:51

Owner

aoh45 (5079)

Last modified by

aoh45 (5079)

Numerical id

4

Author

aoh45 (5079)

Entry type

Example

Classification

msc 13F10

Generated on Fri Feb 9 17:31:25 2018 by

LaTeXML
principal ideal ring

principal ideal ring

A

commutative ring

R

in which all ideals are principal (

http://planetmath.org/PrincipalIdeal

), i.e. (

http://planetmath.org/Ie

) generated by (

http://planetmath.org/IdealGeneratedBy

) a single ring element, is called a

principal ideal ring

.  If

R

is also an

integral domain

, it is a

principal ideal domain

.

Some well-known principal ideal rings are the ring

ℤ

of

integers

, its

factor rings

ℤ

/

n

⁢

ℤ

, and any

polynomial ring

over a field.

Title

principal ideal ring

Canonical name

PrincipalIdealRing

Date of creation

2013-03-22 14:33:16

Last modified on

2013-03-22 14:33:16

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

7

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 13F10

Classification

msc 13A15

Synonym

principal ring

Related topic

CriterionForCyclicRingsToBePrincipalIdealRings

Generated on Fri Feb 9 17:31:27 2018 by

LaTeXML
alternative definition of Krull valuation

alternative definition of Krull valuation

Let

G

be an abelian totally ordered group, denoted additively. We adjoin to

G

a new

element

∞

such that

g

<

+

∞

, for all

g

∈

G

and we extend the

addition

on

G

∞

=

G

∪

{

+

∞

}

by declaring

g

+

(

+

∞

)

=

(

+

∞

)

+

(

+

∞

)

=

+

∞

.

Definition 1

.

Let

R

be an

unital ring

, a

valuation

of

R

with values in

G

is a

function

from

R

to

G

∞

such that , for all

x

,

y

∈

R

:

1)

v

⁢

(

x

⁢

y

)

=

v

⁢

(

x

)

+

v

⁢

(

y

)

,

2)

v

⁢

(

x

+

y

)

≥

min

⁡

{

v

⁢

(

x

)

,

v

⁢

(

y

)

}

,

3)

v

⁢

(

x

)

=

+

∞

iff

v

⁢

(

x

)

=

0

.

Remarks

a) The condition 1) means that

v

is a homomorhism of

R

∖

{

0

}

with

multiplication

in the group

G

. In particular,

v

⁢

(

1

)

=

0

and

v

⁢

(

-

x

)

=

v

⁢

(

x

)

, for all

x

∈

G

. If

x

is

invertible

then

0

=

v

⁢

(

1

)

=

v

⁢

(

x

⁢

x

-

1

)

=

v

⁢

(

x

)

+

v

⁢

(

x

-

1

)

, so

v

⁢

(

x

-

1

)

=

-

v

⁢

(

x

)

.

b) If 3) is replaced by the condition

v

⁢

(

0

)

=

+

∞

then the set

P

=

v

-

1

⁢

{

+

∞

}

is a

prime ideal

of

R

and

v

is on the

integral domain

R

/

P

.

c) In particular, conditions 1) and 3) that

R

is an integral domain and let

K

be its

quotient field

. There is a unique valuation of

K

with values in

G

that extends

v

, namely

v

⁢

(

x

/

y

)

=

v

⁢

(

x

)

-

v

⁢

(

y

)

, for all

x

∈

R

and

y

∈

R

∖

{

0

}

.

d) The element

v

⁢

(

x

)

is sometimes denoted by

v

⁢

x

.

Title

alternative definition of Krull valuation

Canonical name

AlternativeDefinitionOfKrullValuation

Date of creation

2013-03-22 17:02:08

Last modified on

2013-03-22 17:02:08

Owner

polarbear (3475)

Last modified by

polarbear (3475)

Numerical id

10

Author

polarbear (3475)

Entry type

Definition

Classification

msc 13F30

Classification

msc 13A18

Classification

msc 12J20

Classification

msc 11R99

Related topic

OrderValuation

Related topic

Valuation

Related topic

Krullvaluation

Related topic

KrullValuation

Generated on Fri Feb 9 17:31:35 2018 by

LaTeXML
discrete valuation

discrete valuation

A

discrete valuation

on a field

K

is a

valuation

|

⋅

|

:

K

→

ℝ

whose image is a discrete subset of

ℝ

.

For any field

K

with a discrete valuation

|

⋅

|

, the set

R

:=

{

x

∈

K

:

|

x

|

≤

1

}

is a

subring

of

K

with sole

maximal ideal

M

:=

{

x

∈

K

:

|

x

|

<

1

}

,

and hence

R

is a

discrete valuation ring

. Conversely, given any discrete valuation ring

R

, the

field of fractions

K

of

R

admits a discrete valuation sending each element

x

∈

R

to

c

n

, where

0

<

c

<

1

is some arbitrary fixed constant and

n

is the order of

x

, and extending multiplicatively to

K

.

Note:

Discrete valuations are often written additively instead of multiplicatively; under this alternate viewpoint, the element

x

maps to

log

c

⁡

|

x

|

(in the above notation) instead of just

|

x

|

. This transformation reverses the order of the

absolute values

(since

c

<

1

), and sends the element

0

∈

K

to

∞

. It has the advantage that every valuation can be normalized by a suitable scalar multiple to take values in the

integers

.

Title

discrete valuation

Canonical name

DiscreteValuation

Date of creation

2013-03-22 13:59:14

Last modified on

2013-03-22 13:59:14

Owner

djao (24)

Last modified by

djao (24)

Numerical id

6

Author

djao (24)

Entry type

Definition

Classification

msc 13F30

Classification

msc 12J20

Synonym

rank one valuations

Synonym

rank-one valuations

Related topic

DiscreteValuationRing

Related topic

Valuation

Generated on Fri Feb 9 17:31:37 2018 by

LaTeXML
discrete valuation ring

discrete valuation ring

A

discrete valuation ring

R

is a

principal ideal domain

with exactly one

nonzero

maximal ideal

M

. Any

generator

t

of

M

is called a

uniformizer

or

uniformizing element

of

R

; in other words, a uniformizer of

R

is an element

t

∈

R

such that

t

∈

M

but

t

∉

M

2

.

Given a discrete valuation ring

R

and a uniformizer

t

∈

R

, every element

z

∈

R

can be written uniquely in the form

u

⋅

t

n

for some unit

u

∈

R

and some

nonnegative integer

n

∈

ℤ

. The

integer

n

is called the

order

of

z

, and its value is independent of the choice of uniformizing element

t

∈

R

.

Title

discrete valuation ring

Canonical name

DiscreteValuationRing

Date of creation

2013-03-22 12:16:40

Last modified on

2013-03-22 12:16:40

Owner

djao (24)

Last modified by

djao (24)

Numerical id

9

Author

djao (24)

Entry type

Definition

Classification

msc 13F30

Classification

msc 13H10

Synonym

DVR

Related topic

LocalRing

Related topic

DiscreteValuation

Related topic

Valuation

Defines

uniformizer

Defines

uniformizing element

Defines

order

Generated on Fri Feb 9 17:31:39 2018 by

LaTeXML
exponent valuation

exponent valuation

Definition

.  A function

ν

defined in a field

K

is called an

exponent valuation

or shortly an

exponent

of the field, if it satisfies the following conditions:

1.

ν

⁢

(

0

)

=

∞

and

ν

⁢

(

α

)

runs all

rational integers

when

α

runs the nonzero elements of

K

.

2.

ν

⁢

(

α

⁢

β

)

=

ν

⁢

(

α

)

+

ν

⁢

(

β

)

.

3.

ν

⁢

(

α

+

β

)

≧

min

⁡

{

ν

⁢

(

α

)

,

ν

⁢

(

β

)

}

.

Note that because of the discrete value set

ℤ

, an exponent valuation belongs to the

discrete valuations

, and
because of notational causes, to the

order valuations

.

Properties

.

ν

⁢

(

1

)

=

0

ν

⁢

(

-

α

)

=

ν

⁢

(

α

)

ν

⁢

(

α

β

)

=

ν

⁢

(

α

)

-

ν

⁢

(

β

)

ν

⁢

(

α

n

)

=

n

⁢

ν

⁢

(

α

)

ν

⁢

(

α

1

+

…

+

α

n

)

≧

min

⁡

{

ν

⁢

(

α

)

,

…

,

ν

⁢

(

α

n

)

}

ν

⁢

(

α

+

β

)

=

min

⁡

{

ν

⁢

(

α

)

,

ν

⁢

(

β

)

}

if

⁢

ν

⁢

(

α

)

≠

ν

⁢

(

β

)

Example.

If an

integral domain

𝒪

has a

divisor theory

𝒪

*

→

𝔇

, then for each

prime divisor

𝔭

there is an exponent valuation

ν

𝔭

of the

quotient field

K

of

𝒪

.  It is given by

ν

𝔭

(

α

)

=

:

{

∞

when

⁢

α

=

0

,

max

⁡

{

k

∈

ℤ

⁢

⋮

⁢

𝔭

k

∣

(

α

)

}

⁢

when

⁢

α

≠

0

;

ν

𝔭

(

ξ

)

=

:

ν

𝔭

(

α

)

-

ν

𝔭

(

β

)

when

ξ

=

α

β

with

α

,

β

∈

𝒪

*

.

Hence,

𝔭

ν

𝔭

⁢

(

α

)

exactly divides

α

.  Apparently,

ν

𝔭

⁢

(

ξ

)

does not depend on the

quotient

form

α

β

for

ξ

.  It is not hard to show that

ν

𝔭

defined above is an exponent of the field

K

.

Different prime divisors

𝔭

and

𝔮

determine different exponents

ν

𝔭

and

ν

𝔮

, since the condition 3 of the definition of divisor theory (

http://planetmath.org/DivisorTheory

) guarantees such an element

γ

of

𝒪

which in

divisible

by

𝔭

but not by

𝔮

; then

ν

𝔭

⁢

(

γ

)

≧

1

,

ν

𝔮

⁢

(

γ

)

=

0

.

Theorem

.

Let

ν

1

,

…

,

ν

r

be different exponents of a field

K

.  Then for arbitrary set

n

1

,

…

,

n

r

of

integers

, there exists in

K

an element

ξ

such that

ν

1

⁢

(

ξ

)

=

n

1

,

…

,

ν

r

⁢

(

ξ

)

=

n

r

.

The proof of this theorem is found in [1].

References

1

S. Borewicz & I. Safarevic

:

Zahlentheorie

.  Birkhäuser Verlag. Basel und Stuttgart (1966).

Title

exponent valuation

Canonical

name

ExponentValuation

Date of creation

2013-03-22 17:59:31

Last modified on

2013-03-22 17:59:31

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

12

Author

pahio (2872)

Entry type

Definition

Classification

msc 13F30

Classification

msc 13A18

Classification

msc 12J20

Classification

msc 11R99

Synonym

exponent of field

Related topic

DiscreteValuation

Related topic

OrderValuation

Related topic

UltrametricTriangleInequality

Related topic

DivisorTheoryAndExponentValuations

Related topic

DivisorTheory

Defines

exponent of a field

Defines

exponent of the field

Generated on Fri Feb 9 17:31:42 2018 by

LaTeXML
extension of Krull valuation

extension of Krull valuation

The

Krull valuation

|

⋅

|

K

of the field

K

is called the

of the Krull valuation

|

⋅

|

k

of the field

k

, if

k

is a

subfield

of

K

and

|

⋅

|

k

is the restriction of

|

⋅

|

K

on

k

.

Theorem 1

.

The

trivial valuation

is the only of the trivial valuation of

k

to an

algebraic extension

field

K

of

k

.

Proof.

Let’s denote by

|

⋅

|

the trivial valuation of

k

and also its arbitrary Krull to

K

.  Suppose that there is an element

α

of

K

such that

|

α

|

>

1

.  This element satisfies an

algebraic equation

α

n

+

a

1

⁢

α

n

-

1

+

…

+

a

n

=

0

,

where

a

1

,

…

,

a

n

∈

k

.  Since

|

a

j

|

≦

1

for all

j

’s, we get the impossibility

0

=

|

α

n

+

a

1

⁢

α

n

-

1

+

…

+

a

n

|

=

max

⁡

{

|

α

|

n

,

|

a

1

|

⋅

|

α

|

n

-

1

,

…

,

|

a

n

|

}

=

|

α

|

n

>

1

(cf. the sharpening of the

ultrametric triangle inequality

).  Therefore we must have

|

ξ

|

≦

1

for all

ξ

∈

K

,  and because the condition

0

<

|

ξ

|

<

1

would imply that

|

ξ

-

1

|

>

1

,  we see that

|

ξ

|

=

1

∀

ξ

∈

K

∖

{

0

}

,

which that the

valuation

is trivial.

The proof (in [1]) of the next “

extension theorem

” is much longer (one must utilize the extension theorem concerning the

place of field

):

Theorem 2

.

Every Krull valuation of a field

k

can be extended to a Krull valuation of any field of

k

.

References

[1] Emil Artin: .   Lecture notes.  Mathematisches Institut, Göttingen (1959).

Title

extension of Krull valuation

Canonical name

ExtensionOfKrullValuation

Date of creation

2013-03-22 14:55:57

Last modified on

2013-03-22 14:55:57

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

13

Author

pahio (2872)

Entry type

Theorem

Classification

msc 13F30

Classification

msc 13A18

Classification

msc 12J20

Classification

msc 11R99

Related topic

GelfandTornheimTheorem

Generated on Fri Feb 9 17:31:44 2018 by

LaTeXML
extension of valuation from complete base field

extension of valuation from complete base field

Here the

valuations

are of rank one, and it may be supposed that the values are

real numbers

.

•

Assume a

finite field extension

K

/

k

and a valuation of

K

.  If the

base field

is

complete

(

http://planetmath.org/Complete

) with regard to this valuation, so is also the

extension field

.

•

If

K

/

k

is an

algebraic field extension

and if the base field

k

is complete (

http://planetmath.org/Complete

) with regard to its valuation

|

⋅

|

,   then this valuation has one and only one

extension

to the field

K

.  This extension is determined by

|

α

|

=

|

N

⁢

(

α

)

|

n

(

α

∈

K

)

,

where

N

⁢

(

α

)

is the norm of the

element

α

in the

simple field extension

k

⁢

(

α

)

/

k

and

n

is the degree of this field extension.

These

theorems

concern also

Archimedean valuations

.

Title

extension of valuation from complete base field

Canonical name

ExtensionOfValuationFromCompleteBaseField

Date of creation

2013-03-22 15:01:01

Last modified on

2013-03-22 15:01:01

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Theorem

Classification

msc 13F30

Classification

msc 13A18

Classification

msc 12J20

Classification

msc 11R99

Related topic

CompleteUltrametricField

Related topic

ValueGroupOfCompletion

Related topic

NthRoot

Generated on Fri Feb 9 17:31:46 2018 by

LaTeXML
henselian field

henselian field

Let

|

⋅

|

be a

non-archimedean valuation

on a field

K

. Let

V

=

{

x

:

|

x

|

≤

1

}

. Since

|

⋅

|

is ultrametric,

V

is

closed under

addition

and in fact an

additive group

. The other

valuation

axioms
ensure that

V

is a ring. We call

V

the

valuation ring

of

K

with respect to the valuation

|

⋅

|

. Note that the

field of
fractions

of

V

is

K

.

The set

μ

=

{

x

:

|

x

|

<

1

}

is a

maximal ideal

of

V

. The factor

R

:=

V

/

μ

is called the

residue field

or the

residue
class field

.

The map

res

:

V

→

V

/

μ

given by

x

↦

x

+

μ

is called the

residue map

. We extend the definition of the residue map to

sequences

of

elements

from

V

, and hence to

V

⁢

[

X

]

so that if

f

⁢

(

X

)

∈

V

⁢

[

X

]

is given by

∑

i

≤

n

a

i

⁢

X

i

then

res

⁡

(

f

)

∈

R

⁢

[

X

]

is given by

∑

i

≤

n

res

⁡

(

a

⁢

i

)

⁢

X

i

.

Hensel property:

Let

f

⁢

(

x

)

∈

V

⁢

[

x

]

. Suppose

res

⁡

(

f

)

⁢

(

x

)

has a simple root

e

∈

k

. Then

f

⁢

(

x

)

has a root

e

′

∈

V

and

res

⁡

(

e

′

)

=

e

.

Any valued field satisfying the Hensel property is called

henselian

. The

completion

of a non-archimedean valued field

K

with respect to the valuation (cf. constructing the reals from the

rationals

as the completion with respect to the standard metric) is a
henselian field.

Every non-archimedean valued field

K

has a unique (up to

isomorphism

) smallest henselian field

K

h

containing it. We call

K

h

the

henselisation

of

K

.

Title

henselian field

Canonical name

HenselianField

Date of creation

2013-03-22 13:28:37

Last modified on

2013-03-22 13:28:37

Owner

mps (409)

Last modified by

mps (409)

Numerical id

9

Author

mps (409)

Entry type

Definition

Classification

msc 13F30

Classification

msc 13A18

Classification

msc 11R99

Classification

msc 12J20

Related topic

Valuation

Related topic

ValuationDomainIsLocal

Related topic

ValuationRingOfAField

Defines

valuation ring

Defines

residue field

Defines

residue class field

Defines

Hensel property

Defines

henselian

Defines

henselisation

Generated on Fri Feb 9 17:31:49 2018 by

LaTeXML