aarjaneiro/planetmath huggingface dataset:
alternative definition of valuation

alternative definition of valuation

One may use a slightly different version of the third criterion to define a slightly more general

definition

of

valuation

.

A

valuation

on a field

𝕂

is a map

|

⋅

|

:

𝕂

→

ℝ

such that

1.

|

x

|

=

0

if an only if

x

=

0

2.

|

x

⁢

y

|

=

|

x

|

⁢

|

y

|

3.

|

x

+

y

|

≤

C

⁢

max

⁡

{

|

x

|

,

|

y

|

}

The quantity

C

which appears in the third criterion is a positive real number which is known as the .

There is some flexibility in the choice of the

constant

C

in this definition — one can replace

C

by a larger number

C

′

. To deal with this ambiguity, one defines the of the valuation as

inf

⁡

{

C

∣

(

∀

x

)

⁢

(

∀

y

)

⁢

|

x

+

y

|

<

C

⁢

max

⁡

{

|

x

|

,

|

y

|

}

}

The

relation

of this definition to the usual one is the following. On the one hand, if

|

⋅

|

satisfies

the usual definition, then

|

x

+

y

|

≤

|

x

|

+

|

y

|

≤

2

⁢

max

⁡

{

|

x

|

,

|

y

|

}

so a valuation in the old sense is a valuation in the new sense with constant 2.

On the other hand, suppose that

|

⋅

|

satisfies the alternative definition with constant

C

<

2

. Then we have the following result.

Theorem

If

|

⋅

|

is a valuation according to the definition of this entry with constant

C

≤

2

, then

|

⋅

|

satisfies the

identity

|

x

+

y

|

≤

|

x

|

+

|

y

|

.

The proof of this assertion is given in a supplement to this entry.

The foregoing discussion shows that the new definition is more general than the old definition precisely when

C

>

2

. However, this extra generalty is not as great as it might seem at first sight. As is

obvious

from examining the definition, if

|

⋅

|

is a valuation, then so is

|

⋅

|

p

for any power

p

>

0

. Furthermore, if the valuation has constant

C

, then valuation

|

⋅

|

p

has constant

C

p

. Therefore, given any valuation

|

⋅

|

in the sense of this entry, there will exist a number

p

such that

|

⋅

|

p

is a valuation in the sense of the parent entry. Moreover, given the fact that two valuations which are powers of each other are

equivalent

, one sees that the extra generality is not that interesting since the new valuations are equivalent to the old valuations.

Title

alternative definition of valuation

Canonical

name

AlternativeDefinitionOfValuation

Date of creation

2013-03-22 14:55:47

Last modified on

2013-03-22 14:55:47

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

7

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 11R99

Classification

msc 12J20

Classification

msc 13A18

Classification

msc 13F30

Generated on Fri Feb 9 15:46:01 2018 by

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independence of p-adic valuations

independence of

p

-adic valuations

We prove the following particular case:

Proposition 1

.

Let

p

1

,

…

,

p

n

∈

Z

be distinct

prime numbers

and let

∣

⋅

∣

p

i

be the corresponding

p

-adic

valuations

of

Q

. Let

a

1

,

…

,

a

n

∈

Z

and let

ϵ

i

be arbitrary positive real numbers, then there exists

y

∈

Z

such that for all

i

=

1

,

…

,

n

:

∣

y

-

a

i

∣

p

i

<

ϵ

i

Proof.

Let

p

be an arbitrary prime, and let

ϵ

be an arbitrary positive real number. Notice that

ℤ

injects into

ℤ

p

=

lim

←

⁡

ℤ

/

p

n

⁢

ℤ

, the

p

-adic

integers

. For any

b

∈

ℤ

, we also write

b

for its image in

ℤ

p

, and it can be written as a

sequence

b

=

(

b

j

)

with

b

≡

b

j

mod

p

j

. Let

n

=

n

p

,

ϵ

∈

ℕ

be such that

p

-

n

<

ϵ

(and thus for any other

c

∈

ℤ

such that

c

≡

b

n

mod

p

n

we have

∣

b

-

c

∣

p

≤

p

-

n

<

ϵ

).

Now, for the proof of the

proposition

, let

n

i

=

n

p

i

,

ϵ

i

and recall that by the

Chinese Remainder Theorem

we have an

isomorphism

:

∏

i

=

1

n

ℤ

/

p

i

n

i

⁢

ℤ

≡

ℤ

/

(

∏

p

i

n

i

)

⁢

ℤ

Therefore we can find an

element

y

~

of

ℤ

/

(

∏

p

i

n

i

)

⁢

ℤ

(and thus a lift

y

of

y

~

to

ℤ

) such that

y

≡

a

i

mod

p

i

n

i

for all

i

=

1

,

…

,

n

. Hence:

∣

y

-

a

i

∣

p

i

<

ϵ

i

∎

Title

independence of

p

-adic valuations

Canonical name

IndependenceOfPadicValuations

Date of creation

2013-03-22 14:12:14

Last modified on

2013-03-22 14:12:14

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Corollary

Classification

msc 11R99

Related topic

Valuation

Related topic

PAdicIntegers

Related topic

PAdicValuation

Generated on Fri Feb 9 15:46:07 2018 by

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independence of valuations

independence of valuations

Let

|

⋅

|

1

, …,

|

⋅

|

n

be

non-trivial

(i.e., they all have also other values than 0 and 1) and pairwise non-equivalent

valuations

of a field

K

, all with values

real numbers

.  If

a

1

, …,

a

n

are some elements of this field and

ε

is an arbitrary positive number, then there exists in

K

an element

y

which satisfies the conditions

{

|

y

-

a

1

|

1

<

ε

,

⋮

|

y

-

a

n

|

n

<

ε

.

Title

independence of valuations

Canonical name

IndependenceOfValuations

Date of creation

2013-03-22 14:11:44

Last modified on

2013-03-22 14:11:44

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

22

Author

pahio (2872)

Entry type

Theorem

Classification

msc 11R99

Synonym

approximation theorem

Related topic

TrivialValuation

Related topic

EquivalentValuations

Related topic

WeakApproximationTheorem

Generated on Fri Feb 9 15:46:09 2018 by

LaTeXML
examples of ramification of archimedean places

examples of ramification of archimedean places

Example 1

.

Let

K

=

ℚ

⁢

(

-

d

)

be a

quadratic imaginary number field

. Then

K

has only two

embeddings

which, in fact, are complex-conjugate embeddings:

ψ

:

K

→

ℂ

,

-

d

→

-

d

ψ

¯

:

K

→

ℂ

,

-

d

→

-

-

d

The

archimedean place

w

=

(

ψ

,

ψ

¯

)

is lying above the unique archimedean place of

ℚ

:

ϕ

:

ℚ

→

ℝ

and therefore, the place

v

=

ϕ

ramifies in

K

.

Example 2

.

Let

K

be a CM-field i.e.

K

is a totally imaginary (

http://planetmath.org/TotallyRealAndImaginaryFields

)

quadratic extension

of a totally real field

K

+

. Then we claim that the

extension

K

/

K

+

is

totally ramified

at the

archimedean

(or

infinite

) places. Indeed, let

v

be an archimedean place of

K

+

. By

assumption

,

K

+

is a totally real field, thus all its places are real, and so,

v

is real. Let

w

be any archimedean place of

K

lying above

v

(i.e. extending

v

to

K

). Since

K

is totally imaginary, the place

w

is a pair of

complex embeddings

, and therefore

v

ramifies in

K

/

K

+

. Thus, all archimedean places of

K

+

ramify in

K

and

e

(

w

|

v

)

=

2

for all

w

|

v

.

Title

examples of ramification of archimedean places

Canonical name

ExamplesOfRamificationOfArchimedeanPlaces

Date of creation

2013-03-22 15:07:29

Last modified on

2013-03-22 15:07:29

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Example

Classification

msc 11S15

Classification

msc 13B02

Classification

msc 12F99

Related topic

TotallyRealAndImaginaryFields

Related topic

ExamplesOfPrimeIdealDecompositionInNumberFields

Generated on Fri Feb 9 15:46:24 2018 by

LaTeXML
unramified action

unramified action

Let

K

be a

number field

and let

ν

be a

discrete valuation

on

K

(this might be, for example, the

valuation

attached to a

prime
ideal

𝔓

of

K

).

Let

K

ν

be the

completion

of

K

at

ν

, and let

𝒪

ν

be the

ring of integers

of

K

ν

, i.e.

𝒪

ν

=

{

k

∈

K

ν

∣

ν

⁢

(

k

)

≥

0

}

The

maximal ideal

of

𝒪

ν

will be denoted by

ℳ

=

{

k

∈

K

ν

∣

ν

⁢

(

k

)

>

0

}

and we denote by

k

ν

the

residue field

of

K

ν

, which
is

k

ν

=

𝒪

ν

/

ℳ

We will consider three different global

Galois groups

, namely

G

K

¯

/

K

=

Gal

⁡

(

K

¯

/

K

)

G

K

ν

¯

/

K

ν

=

Gal

⁡

(

K

ν

¯

/

K

ν

)

G

k

ν

¯

/

k

ν

=

Gal

⁡

(

k

ν

¯

/

k

ν

)

where

K

¯

,

K

ν

¯

,

k

ν

¯

are

algebraic closures

of the corresponding field. We also
define notation for the

inertia group

of

G

K

ν

¯

/

K

ν

I

ν

⊆

G

K

ν

¯

/

K

ν

Definition 1

.

Let

S

be a set and suppose there is a

group action

of

G

⁢

a

⁢

l

⁢

(

K

ν

¯

/

K

ν

)

on

S

. We say that

S

is

unramified

at

ν

, or the action of

G

K

ν

¯

/

K

ν

on

S

is unramified at

ν

, if the action of

I

ν

on

S

is trivial,
i.e.

σ

⁢

(

s

)

=

s

∀

σ

∈

I

ν

,

∀

s

∈

𝒮

Remark

: By

Galois theory

we know that,

K

ν

nr

, the

fixed field

of

I

ν

, the
inertia

subgroup

, is the

maximal

unramified

extension

of

K

ν

, so

I

ν

≅

Gal

⁡

(

K

ν

¯

/

K

ν

nr

)

Title

unramified action

Canonical name

UnramifiedAction

Date of creation

2013-03-22 13:56:26

Last modified on

2013-03-22 13:56:26

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Definition

Classification

msc 11S15

Synonym

set is unramified at a valuation

Related topic

InfiniteGaloisTheory

Related topic

DecompositionGroup

Related topic

Valuation

Generated on Fri Feb 9 15:46:32 2018 by

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fundamental theorem of Galois theory

fundamental theorem of Galois theory

Let

L

/

F

be a

Galois extension

of finite degree,
with

Galois group

G

:=

Gal

⁡

(

L

/

F

)

.
There is a

bijective

, inclusion-reversing correspondence
between

subgroups

of

G

and

extensions

of

F

contained

in

L

, given by

•

K

↦

Gal

⁡

(

L

/

K

)

,
for any field

K

with

F

⊆

K

⊆

L

.

•

H

↦

L

H

(the

fixed field

of

H

in

L

),
for any subgroup

H

≤

G

.

The extension

L

H

/

F

is normal if and only if

H

is a

normal subgroup

of

G

,
and in this case the

homomorphism

G

⟶

Gal

⁡

(

L

H

/

F

)

given by

σ

↦

σ

|

L

H

induces (via the

first isomorphism theorem

)
a natural identification

Gal

⁡

(

L

H

/

F

)

=

G

/

H

between the Galois group of

L

H

/

F

and the

quotient group

G

/

H

.

For the case of Galois extensions of

infinite

degree,
see the entry on

infinite Galois theory

.

Title

fundamental theorem of Galois theory

Canonical name

FundamentalTheoremOfGaloisTheory

Date of creation

2013-03-22 12:08:31

Last modified on

2013-03-22 12:08:31

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

9

Author

yark (2760)

Entry type

Theorem

Classification

msc 11S20

Classification

msc 11R32

Classification

msc 12F10

Classification

msc 13B05

Synonym

Galois theory

Synonym

Galois correspondence

Related topic

GaloisTheoreticDerivationOfTheCubicFormula

Related topic

GaloisTheoreticDerivationOfTheQuarticFormula

Related topic

InfiniteGaloisTheory

Related topic

GaloisGroup

Generated on Fri Feb 9 15:46:36 2018 by

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Hilbert symbol

Hilbert symbol

Let

K

be any

local field

. For any two nonzero elements

a

,

b

∈

K

×

, we define:

(

a

,

b

)

:=

{

+

1

if

z

2

=

a

⁢

x

2

+

b

⁢

y

2

has a nonzero solution

(

x

,

y

,

z

)

≠

(

0

,

0

,

0

)

in

K

3

,

-

1

otherwise.

The number

(

a

,

b

)

is called the

Hilbert symbol

of

a

and

b

in

K

.

Title

Hilbert symbol

Canonical

name

HilbertSymbol

Date of creation

2013-03-22 12:50:48

Last modified on

2013-03-22 12:50:48

Owner

djao (24)

Last modified by

djao (24)

Numerical id

5

Author

djao (24)

Entry type

Definition

Classification

msc 11S31

Classification

msc 11S80

Generated on Fri Feb 9 15:46:38 2018 by

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Mahler’s theorem for continuous functions on the p-adic integers

Mahler’s theorem for continuous functions on the

p

-adic integers

Theorem

.

(Mahler)
Let

f

be a

continuous function

on the

p

-adic integers taking values in some

finite extension

K

of

Q

p

, and for each

n

∈

N

, put

a

n

=

∑

i

=

0

n

(

-

1

)

n

-

i

⁢

(

n

i

)

⁢

f

⁢

(

i

)

. Then

a

n

→

0

as

n

→

∞

, the series

∑

n

=

0

∞

a

n

⁢

(

⋅

n

)

converges uniformly to

f

on

Z

p

, and

∥

f

∥

∞

=

sup

n

≥

0

⁡

|

a

n

|

p

, where

∥

⋅

∥

∞

denotes the sup norm.

Title

Mahler’s

theorem

for continuous functions on the

p

-adic integers

Canonical name

MahlersTheoremForContinuousFunctionsOnThePadicIntegers

Date of creation

2013-03-22 18:32:07

Last modified on

2013-03-22 18:32:07

Owner

azdbacks4234 (14155)

Last modified by

azdbacks4234 (14155)

Numerical id

6

Author

azdbacks4234 (14155)

Entry type

Theorem

Classification

msc 11S80

Generated on Fri Feb 9 15:46:42 2018 by

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p-adic analytic

p-adic analytic

Definition

.

Let

C

p

be the field of

complex

p

-adic numbers (

http://planetmath.org/ComplexPAdicNumbers

). Let

U

be a domain in

C

p

. A

function

f

:

U

⟶

C

p

is

p

-adic

analytic

if

f

has a

Taylor series

(with coefficients in

C

p

) about each point

z

∈

U

that

converges

to the function

f

in an open neighborhood of

z

.

For example, the

p

-adic

exponential function

(

http://planetmath.org/PAdicExponentialAndPAdicLogarithm

) is analytic on its

domain of definition

:

U

=

{

z

∈

ℂ

p

:

|

z

|

p

<

1

p

1

/

(

p

-

1

)

}

.

The study of

p

-adic

analytic functions

is usually called

p

-adic

analysis

and it is very

similar

to

complex analysis

in many respects, although there are important

differences

coming from the distinct

topologies

of

ℂ

and

ℂ

p

.

Title

p-adic analytic

Canonical

name

PadicAnalytic

Date of creation

2013-03-22 15:13:53

Last modified on

2013-03-22 15:13:53

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Definition

Classification

msc 11S99

Classification

msc 12J12

Classification

msc 11S80

Synonym

p

-adic analytic

Related topic

Analytic

Related topic

PAdicExponentialAndPAdicLogarithm

Defines

p

-adic analysis

Defines

p-adic analysis

Generated on Fri Feb 9 15:46:47 2018 by

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p-adic integers

p

-adic integers

1

Basic construction

For any prime

p

, the

p

–adic integers

is the ring obtained by taking the

completion

of the integers

ℤ

with respect to the metric induced by the norm

|

x

|

:=

1

p

ν

p

⁢

(

x

)

,

x

∈

ℤ

,

(1)

where

ν

p

⁢

(

x

)

denotes the largest integer

e

such that

p

e

divides

x

. The induced metric

d

⁢

(

x

,

y

)

:=

|

x

-

y

|

is called the

p

–adic metric

on

ℤ

. The ring of

p

–adic integers is usually denoted by

ℤ

p

, and its

fraction field

by

ℚ

p

.

2

Profinite viewpoint

The ring

ℤ

p

of

p

–adic integers can also be constructed by taking the

inverse limit

ℤ

p

:=

lim

⟵

⁢

ℤ

/

p

n

⁢

ℤ

over the

inverse system

⋯

→

ℤ

/

p

2

⁢

ℤ

→

ℤ

/

p

⁢

ℤ

→

0

consisting of the rings

ℤ

/

p

n

⁢

ℤ

, for all

n

≥

0

, with the projection maps defined to be the unique maps such that the

diagram

Generated on Fri Feb 9 15:46:53 2018 by

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freshman’s dream

freshman’s dream

Theorem 1

(Freshman’s dream).

If

k

if a field of

characteristic

(

http://planetmath.org/Characteristic

)

p

>

0

(so

p

is prime)
then for all

x

,

y

∈

k

we have

(

x

+

y

)

p

i

=

x

p

i

+

y

p

i

.

Therefore

x

↦

x

p

i

is a

field monomorphism

(called a Frobenius

monomorphism

.)

When

k

is finite then it is indeed an automorphism.
A field

k

is called a

perfect field

when the map is

surjective

.

The

theorem

is so named because it is a common mistake for freshman math students
to make over the

real numbers

. However, as the characteristic of the real numbers
is 0, this does not apply in any interesting way to that setting.

It should also be noted that the result applies only to powers of the characteristic,
and not all exponents.

Proof.

The proof is an application of the

binomial theorem

. We prove it for

p

first.

(

x

+

y

)

p

=

∑

i

=

0

p

(

p

i

)

⁢

x

i

⁢

y

p

-

i

.

Now observe

(

p

i

)

=

p

!

(

p

-

i

)

!

⁢

i

!

=

p

⋅

(

p

-

1

)

!

(

p

-

i

)

!

⁢

i

!

.

As

p

is prime and

1

≤

i

≤

p

-

1

it follows

i

!

and

(

p

-

i

)

!

do not divide

p

.
As the field

k

has characteristic

p

,

(

p

-

1

)

!

(

p

-

i

)

!

⁢

i

!

is an

integer

m

where

(

p

i

)

=

p

⁢

m

≡

0

.

Thus

(

x

+

y

)

p

=

x

p

+

y

p

.

Now for

p

i

simply use

induction

:

(

x

+

y

)

p

i

=

(

(

x

+

y

)

p

)

p

i

-

1

=

(

x

p

+

y

p

)

p

i

-

1

=

x

p

i

+

y

p

i

.

∎

Title

freshman’s dream

Canonical name

FreshmansDream

Date of creation

2013-03-22 15:51:17

Last modified on

2013-03-22 15:51:17

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

18

Author

Algeboy (12884)

Entry type

Theorem

Classification

msc 11T23

Classification

msc 11T30

Synonym

Frobenius Automorphism

Related topic

PolynomialCongruence

Generated on Fri Feb 9 15:46:55 2018 by

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binary Golay code

binary Golay code

The

binary Golay Code

𝒢

23

is a

perfect

linear binary [23,12,7]-code with a plethora of different constructions.

Sample Constructions

•

Lexicographic Construction:

Let

v

0

be the all-zero word in

𝔽

2

23

, and inductively define

v

j

to be the smallest word (smallest with respect to the lexicographic ordering on

𝔽

2

23

that differs from

v

i

in at least 7 places for all

i

<

j

.

•

Construction:

𝒢

23

is the quadratic residue code of length 23.

The

extended binary Golay Code

𝒢

24

is obtained by appending a zero-sum check digit to the end of every word in

𝒢

23

.

Both the binary Golay code and the extended binary Golay code have some remarkable .

Properties

•

𝒢

24

has 4096

codewords

: 1 of weight 0, 759 of weight 8, 2576 of weight 12, 759 of weight 18, and 1 of weight 24.

•

The

automorphism group

of

𝒢

24

is the

Mathieu group

M

24

, one of the

sporadic groups

.

•

The

Golay Code

is used to define the

Leech Lattice

, one of the most efficient sphere-packings known to date.

•

The optimal strategy to the mathematical game called Mogul is to always revert the current position to one corresponding to a word of the Golay code.

•

The words of weight 8 in

𝒢

24

form a

S

⁢

(

5

,

8

,

24

)

Steiner system

. In fact, this property uniquely determines the code.

Title

binary Golay code

Canonical name

BinaryGolayCode

Date of creation

2013-03-22 14:23:39

Last modified on

2013-03-22 14:23:39

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

4

Author

mathcam (2727)

Entry type

Definition

Classification

msc 11T71

Related topic

LeechLattice

Related topic

Hexacode

Defines

extended binary golay code

Generated on Fri Feb 9 15:46:57 2018 by

LaTeXML
perfect code

perfect code

Let

C

be a linear (

http://planetmath.org/LinearCode

)

(

n

,

k

,

d

)

-code over

𝔽

q

.

The

packing radius

of

C

is defined to be the value

ρ

⁢

(

C

)

=

d

-

1

2

.

The

covering radius

of

C

is

r

⁢

(

C

)

=

max

x

⁡

min

c

⁡

δ

⁢

(

x

,

c

)

with

x

∈

𝔽

q

n

and

c

∈

C

, and where

δ

denotes the

Hamming distance

on

𝔽

q

n

.

The code (

http://planetmath.org/Code

)

C

is said to be

perfect

if

r

⁢

(

C

)

=

ρ

⁢

(

C

)

.

The list of of linear

perfect codes

is very short, including only trivial codes,

Hamming codes

(i.e.

ρ

=

1

), and the binary and ternary Golay (

http://planetmath.org/BinaryGolayCode

) codes.

Title

perfect code

Canonical

name

PerfectCode

Date of creation

2013-03-22 14:23:43

Last modified on

2013-03-22 14:23:43

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

5

Author

mathcam (2727)

Entry type

Definition

Classification

msc 11T71

Defines

packing radius

Defines

covering radius

Generated on Fri Feb 9 15:46:59 2018 by

LaTeXML
generator for the mutiplicative group of a field

generator for the mutiplicative group of a field

Proposition 1

The

multiplicative group

K

*

of a

finite field

K

is cyclic.

Theorem

3.1 in the finite fields (

http://planetmath.org/FiniteField

) entry proves
this

proposition

along with a more general result:

Proposition 2

If for every

natural number

d

, the

equation

x

d

=

1

has at most

d

solutions in a

finite group

G

then

G

is cyclic. Equivalently, for any

positive

divisor

d

of

|

G

|

.

This last proposition implies that every

finite subgroup

of the multiplicative group of a field (finite or not) is cyclic.

We will give an alternative

constructive proof

of Proposition 1:

We first factorize

q

-

1

=

∏

i

=

1

n

p

i

e

i

. There exists an

element

y

i

in

K

*

such that

y

i

is not root of

x

(

q

-

1

)

/

p

i

-

1

, since the

polynomial

has degree less than

q

-

1

. Let

z

i

=

y

i

(

q

-

1

)

/

p

i

e

i

. We note that

z

i

has order

p

i

e

i

. In fact

z

i

p

i

e

i

=

1

and

z

i

p

i

e

i

-

1

=

y

i

(

q

-

1

)

/

p

i

≠

1

.

Finally we choose the element

z

=

∏

i

=

1

n

z

i

. By the Theorem 1 here (

http://planetmath.org/OrderOfElementsInFiniteGroups

), we obtain that the order of

z

is

q

-

1

i.e.

z

is a

generator

of the

cyclic group

K

*

.

Title

generator for the mutiplicative group of a field

Canonical name

GeneratorForTheMutiplicativeGroupOfAField

Date of creation

2013-03-22 16:53:17

Last modified on

2013-03-22 16:53:17

Owner

polarbear (3475)

Last modified by

polarbear (3475)

Numerical id

16

Author

polarbear (3475)

Entry type

Result

Classification

msc 11T99

Classification

msc 12E20

Generated on Fri Feb 9 15:47:01 2018 by

LaTeXML