aarjaneiro/planetmath huggingface dataset:
equivalent valuations

equivalent valuations

Let

K

be a field.  The

equivalence of valuations

|

⋅

|

1

and

|

⋅

|

2

of

K

may be defined so that

1.

|

⋅

|

1

is not the

trivial valuation

;

2.

if

|

a

|

1

<

1

then

|

a

|

2

<

1

∀

a

∈

K

.

It it easy to see that these conditions imply for both

valuations

(use

1

a

).  Also, we have always

|

a

|

1

≦

1

⇔

|

a

|

2

≦

1

;

so both valuations have a common

valuation ring

in the case they are non-archimedean.  (The of the more general

Krull valuations

is defined to that they have common valuation rings.)  Further, both valuations determine a common metric on

K

.

Theorem

.

Two valuations (of rank (

http://planetmath.org/KrullValuation

) one)

|

⋅

|

1

and

|

⋅

|

2

of

K

are iff one of them is a

positive

power of the other,

|

a

|

1

=

|

a

|

2

c

∀

a

∈

K

,

where

c

is a positive .

Title

equivalent valuations

Canonical name

EquivalentValuations

Date of creation

2013-03-22 14:25:27

Last modified on

2013-03-22 14:25:27

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

18

Author

pahio (2872)

Entry type

Definition

Classification

msc 13A18

Related topic

DiscreteValuation

Related topic

IndependenceOfTheValuations

Defines

equivalence of valuations

Generated on Fri Feb 9 17:23:00 2018 by

LaTeXML
Ostrowski’s valuation theorem

Ostrowski’s valuation theorem

The

field of rational

numbers has no other non-equivalent (

http://planetmath.org/EquivalentValuations

)

valuations

than

•

the

trivial valuation

,

•

the

absolute value

, i.e. the

complex modulus

|

⋅

|

∞

and

•

the

p

-adic valuations

|

⋅

|

p

when

p

goes through all

positive

primes.

Note.

Any valuation

|

⋅

|

of the field

ℚ

defines a metric

d

⁢

(

x

,

y

)

=

|

x

-

y

|

in the field, but

ℚ

is

complete

(

http://planetmath.org/Complete

) only with respect to (the “trivial metric” defined by) the trivial valuation.  The field has the proper

completions

with respect to its other valuations:  the

field of reals

ℝ

and the fields

ℚ

p

of

p

-adic numbers (

http://planetmath.org/PAdicIntegers

); cf. also

p

-adic

canonical form

(

http://planetmath.org/PAdicCanonicalForm

).

Title

Ostrowski’s valuation theorem

Canonical name

OstrowskisValuationTheorem

Date of creation

2013-03-22 14:55:30

Last modified on

2013-03-22 14:55:30

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Theorem

Classification

msc 13A18

Generated on Fri Feb 9 17:23:02 2018 by

LaTeXML
p-adic valuation

p-adic valuation

Let

p

be a

positive

prime number

.  For every non-zero

rational number

x

there exists a unique

integer

n

such that

x

=

p

n

⋅

u

v

with some integers

u

and

v

indivisible by

p

.  We define

|

x

|

p

:=

{

(

1

p

)

n

when

⁢

x

≠

0

,

0

when

⁢

x

=

0

,

obtaining a non-trivial (

http://

planetmath

.org/TrivialValuation

)

non-archimedean valuation

, the so-called

p

-

adic

valuation

|

⋅

|

p

:

ℚ

→

ℝ

of the field

ℚ

.

The

value group

of the

p

-adic valuation consists of all integer-powers of the prime number

p

.  The

valuation ring

of the valuation is called the ring of the

p-integral rational numbers

; their

denominators

, when

reduced

(

http://planetmath.org/Fraction

) to lowest terms, are not

divisible

by

p

.

The

field of rationals

has the

2-adic, 3-adic, 5-adic, 7-adic

and so on valuations (which may be called, according to Greek,

dyadic, triadic, pentadic, heptadic

and so on).  They all are non-equivalent (

http://planetmath.org/EquivalentValuations

) with each other.

If one replaces the number

1

p

by any positive

ϱ

less than 1, one obtains an

equivalent

(

http://planetmath.org/EquivalentValuations

)

p

-adic valuation; among these the valuation with

ϱ

=

1

p

is sometimes called the

normed

p

-adic valuation

.  Analogously we can say that the

absolute value

is the

normed archimedean valuation

of

ℚ

which corresponds the infinite prime

∞

of

ℤ

.

The

product

of all normed valuations of

ℚ

is the

trivial valuation

|

⋅

|

tr

,  i.e.

∏

p

⁢

prime

|

x

|

p

=

|

x

|

tr

∀

x

∈

ℚ

.

Title

p-adic valuation

Canonical name

PadicValuation

Date of creation

2013-03-22 14:55:50

Last modified on

2013-03-22 14:55:50

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

14

Author

pahio (2872)

Entry type

Definition

Classification

msc 13A18

Synonym

p

-adic valuation

Related topic

IndependenceOfPAdicValuations

Related topic

IntegralElement

Related topic

OrderValuation

Related topic

StrictDivisibility

Defines

p

-

integral

rational number

Defines

normed

p

-adic valuation

Defines

normed archimedean valuation

Defines

dyadic valuation

Defines

triadic valuation

Defines

pentadic valuation

Defines

heptadic valuation

Generated on Fri Feb 9 17:23:06 2018 by

LaTeXML
place as extension of homomorphism

place as extension of homomorphism

Theorem

.

If

f

is a

ring homomorphism

from a

subring

𝔬

of a field

k

to an algebraically closed field

F

such that

f

⁢

(

1

)

=

1

,  then there exists a place (

http://planetmath.org/PlaceOfField

)

φ

:

k

→

F

∪

{

∞

}

of the field

k

such that

φ

|

𝔬

=

f

.

Note.

That

F

should be

algebraically closed

, does not , since every field is extendable to such one.

References

1

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

Title

place as extension of homomorphism

Canonical name

PlaceAsExtensionOfHomomorphism

Date of creation

2013-03-22 14:57:21

Last modified on

2013-03-22 14:57:21

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Theorem

Classification

msc 13A18

Classification

msc 12E99

Classification

msc 13F30

Synonym

extension theorem

Related topic

RamificationOfArchimedeanPlaces

Generated on Fri Feb 9 17:23:09 2018 by

LaTeXML
proof of theorem on equivalent valuations

proof of theorem on equivalent valuations

It is easy to see that

|

⋅

|

and

|

⋅

|

c

are

equivalent valuations

for any

constant

c

>

0

— it follows from the fact that

0

≤

x

c

<

1

if and only if

0

<

x

≤

1

.

Assume that the

valuations

|

⋅

|

1

and

|

⋅

|

2

are

equivalent

.
Let

b

be an element of

K

such that

0

<

|

b

|

1

<

1

. Because the valuations are assumed to be equivalent, it is also the case that

0

<

|

b

|

2

<

1

. Hence, there must exist positive constants

c

1

and

c

2

such that

|

b

|

1

c

1

=

1

2

and

|

b

|

2

c

2

=

1

2

.

We will show that show that

|

x

|

1

c

1

=

|

x

|

2

c

2

for all

a

∈

K

by

contradiction

.

Let

a

be any element of

k

such that

0

<

|

a

|

1

<

1

. Assume that

|

a

|

1

c

1

≠

|

a

|

2

c

2

. Then either

|

a

|

1

c

1

<

|

a

|

2

c

2

or

|

a

|

1

c

1

>

|

a

|

2

c

2

. We may assume that

|

a

|

1

c

1

<

|

a

|

2

c

2

without loss of generality.

Since

|

a

|

2

c

2

/

|

a

|

1

c

1

>

1

, there exists an integer

m

>

0

such that

(

|

a

|

2

c

2

/

|

a

|

1

c

1

)

m

>

2

. Let

n

be the least integer such that

2

n

⁢

|

a

|

2

m

⁢

c

2

>

1

. Then we have

2

n

⁢

|

a

|

1

m

⁢

c

1

⁢

<

2

n

-

1

|

⁢

a

|

2

m

⁢

c

2

<

1

⁢

<

2

n

|

⁢

a

|

2

m

⁢

c

2

.

Since

2

=

|

b

-

1

|

1

c

1

=

|

b

-

1

|

2

c

2

, this implies that

|

a

m

b

n

|

1

c

1

<

1

<

|

a

m

b

n

|

2

c

2

,

but then

|

a

m

b

n

|

1

<

1

and

|

a

m

b

n

|

2

>

1

,

which is impossible because the two valuations are assumed to be equivalent.

Q.E.D

Title

proof of theorem on equivalent valuations

Canonical name

ProofOfTheoremOnEquivalentValuations

Date of creation

2013-03-22 14:55:40

Last modified on

2013-03-22 14:55:40

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

12

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 13A18

Generated on Fri Feb 9 17:23:20 2018 by

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invariant polynomial

invariant polynomial

An

invariant polynomial

is a

polynomial

P

that is

invariant

under a (compact) Lie group

Γ

acting on a

vector space

V

. Therefore

P

is

Γ

-invariant polynomial if

P

⁢

(

γ

⁢

x

)

=

P

⁢

(

x

)

for all

γ

∈

Γ

and

x

∈

V

.

References

GSS

Golubitsky, Martin. Stewart, Ian. Schaeffer, G. David: Singularities and Groups in Bifurcation Theory

(Volume II)

. Springer-Verlag, New York, 1988.

Title

invariant polynomial

Canonical name

InvariantPolynomial

Date of creation

2013-03-22 13:40:19

Last modified on

2013-03-22 13:40:19

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

6

Author

Daume (40)

Entry type

Definition

Classification

msc 13A50

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

LaTeXML
Schwarz (1975) theorem

Schwarz (1975) theorem

theorem:

Let

Γ

be a

compact Lie group

acting on

V

. Let

u

1

,

…

,

u

s

be a Hilbert basis for the

Γ

-

invariant polynomials

𝒫

⁢

(

Γ

)

(see

Hilbert-Weyl theorem

)

. Let

f

∈

ℰ

⁢

(

Γ

)

. Then there exists a smooth germ

h

∈

ℰ

s

(the ring of

C

∞

germs

R

s

→

R

)

such that

f

⁢

(

x

)

=

h

⁢

(

u

1

⁢

(

x

)

,

…

,

u

s

⁢

(

x

)

)

.

[

GSS

]

proof:

The proof is shown on page 58 of

[

GSS

]

.

theorem:

(as stated by Gerald W. Schwarz)

Let

G

be a compact Lie group acting orthogonally on

ℝ

n

, let

ρ

1

,

…

,

ρ

k

be

generators

of

𝒫

⁢

(

ℝ

n

)

G

(the set

G

-invariant polynomials on

R

n

)

, and let

ρ

=

(

ρ

1

,

…

,

ρ

k

)

:

ℝ

n

→

ℝ

k

. Then

ρ

*

ℰ

⁢

(

ℝ

k

)

=

ℰ

⁢

(

ℝ

n

)

G

.

[

SG

]

proof:

The proof is shown in the following publication

[

SG

]

.

References

GSS

Golubitsky, Martin. Stewart, Ian. Schaeffer, G. David: Singularities and Groups in Bifurcation Theory

(Volume II)

. Springer-Verlag, New York, 1988.

SG

Schwarz, W. Gerald: Smooth Functions Invariant Under the Action of a Compact Lie Group,

Topology

Vol. 14, pp. 63-68, 1975.

Title

Schwarz (1975) theorem

Canonical name

Schwarz1975Theorem

Date of creation

2013-03-22 13:40:06

Last modified on

2013-03-22 13:40:06

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 13A50

Generated on Fri Feb 9 17:23:27 2018 by

LaTeXML
a finite ring is cyclic if and only its order and characteristic are equal

a finite ring is cyclic if and only its order and characteristic are equal

.

A

finite ring

is cyclic if and only if its order (

http://planetmath.org/OrderRing

) and

characteristic

are equal.

Proof.

If

R

is a

cyclic ring

and

r

is a

generator

(

http://planetmath.org/Generator

) of the

additive group

of

R

, then

|

r

|

=

|

R

|

. Since, for every

s

∈

R

,

|

s

|

divides

|

R

|

, then it follows that

char

⁡

R

=

|

R

|

. Conversely, if

R

is a finite ring such that

char

⁡

R

=

|

R

|

, then the

exponent

of the additive group of

R

is also equal to

|

R

|

. Thus, there exists

t

∈

R

such that

|

t

|

=

|

R

|

. Since

⟨

t

⟩

is a

subgroup

of the additive group of

R

and

|

⟨

t

⟩

|

=

|

t

|

=

|

R

|

, it follows that

R

is a cyclic ring.∎

Title

a

finite ring is cyclic if and only its order and characteristic are equal

Canonical name

AFiniteRingIsCyclicIfAndOnlyItsOrderAndCharacteristicAreEqual

Date of creation

2013-03-22 13:30:30

Last modified on

2013-03-22 13:30:30

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

12

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 13A99

Generated on Fri Feb 9 17:23:30 2018 by

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algebra without order

algebra without order

An

algebra

(

http://planetmath.org/Algebra

)

A

is said to be

\PMlinkescapephrase

order

without order

if it is

commutative

and

for each

a

∈

A

, there exists

b

∈

A

such that

a

⁢

b

≠

0

.

The phrase

algebra without order

seems first in the book “Multipliers of Banach algebras” by Ronald Larsen. In

noncommutative

case, the

concept

is divied into two parts – without left/right order. However, in the noncommutative case, it is defined in terms of the injectivity of the left (right)

regular representation

given by

x

∈

A

↦

L

x

∈

L

⁢

(

A

)

.

Note that for an algebra

A

and an

element

x

∈

A

,

L

x

:

A

→

A

is the map defined by

L

x

⁢

(

y

)

=

x

⁢

y

. Then

L

x

is a linear operator on

A

. It is easy to see that

A

is without left order if and only if the map

x

∈

A

↦

L

x

∈

L

⁢

(

A

)

is one-one; equivalently, the

left ideal

{

x

∈

A

:

x

∈

A

}

=

{

0

}

. This ideal is is called the

left annihilator

of

A

.

Every

commutative algebra

with

identity

is without order.

Example:

ℝ

2

with

multiplication

defined by

(

x

1

,

x

2

)

*

(

y

1

,

y

2

)

=

(

x

1

⁢

y

1

,

0

)

, (

(

x

1

,

x

2

)

,

(

y

1

,

y

2

)

∈

ℝ

2

) is not an algebra without order as multiplication of (0,1) with any other element gives (0,0).

Title

algebra without order

Canonical name

AlgebraWithoutOrder

Date of creation

2013-03-22 14:46:01

Last modified on

2013-03-22 14:46:01

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

11

Author

mathcam (2727)

Entry type

Definition

Classification

msc 13A99

Generated on Fri Feb 9 17:23:34 2018 by

LaTeXML
arithmetical ring

arithmetical ring

Theorem

.

If

R

is a

commutative ring

, then the following three conditions are

equivalent

:

•

For all ideals

𝔞

,

𝔟

and

𝔠

of

R

, one has

𝔞

∩

(

𝔟

+

𝔠

)

=

(

𝔞

∩

𝔟

)

+

(

𝔞

∩

𝔠

)

.

•

For all ideals

𝔞

,

𝔟

and

𝔠

of

R

, one has

𝔞

+

(

𝔟

∩

𝔠

)

=

(

𝔞

+

𝔟

)

∩

(

𝔞

+

𝔠

)

.

•

For each

maximal ideal

𝔭

of

R

the set of all ideals of

R

𝔭

, the localisation (

http://planetmath.org/

Localization

) of

R

at

R

∖

𝔭

,  is

totally ordered

by

set inclusion

.

The ring

R

satisfying the conditions of the

theorem

is called an

arithmetical ring

.

Title

arithmetical ring

Canonical

name

ArithmeticalRing

Date of creation

2013-03-22 15:23:58

Last modified on

2013-03-22 15:23:58

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

8

Author

PrimeFan (13766)

Entry type

Theorem

Classification

msc 13A99

Related topic

QuotientOfIdeals

Generated on Fri Feb 9 17:23:37 2018 by

LaTeXML
behavior

behavior

If

R

is an

infinite cyclic

ring (

http://planetmath.org/CyclicRing3

), the

behavior

of

R

is a

nonnegative integer

k

such that there exists a

generator

(

http://planetmath.org/Generator

)

r

of the

additive group

of

R

with

r

2

=

k

⁢

r

.

If

R

is a finite

cyclic ring

of order

n

, the

behavior

of

R

is a

positive

divisor

k

of

n

such that there exists a generator

r

of the additive group of

R

with

r

2

=

k

⁢

r

.

For any cyclic ring, behavior exists uniquely. Moreover, the behavior of a cyclic ring determines many of its .

To the best of my knowledge, this definition first appeared in my master’s thesis:

Buck, Warren.

http://planetmath.org/?op=getobj&from=papers&id=336

Cyclic Rings

. Charleston, IL: Eastern Illinois University, 2004.

Title

behavior

Canonical name

Behavior

Date of creation

2013-03-22 16:02:29

Last modified on

2013-03-22 16:02:29

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

15

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 13A99

Classification

msc 16U99

Generated on Fri Feb 9 17:23:40 2018 by

LaTeXML
behavior exists uniquely (infinite case)

behavior exists uniquely (infinite case)

The following is a proof that behavior exists uniquely for any

infinite cyclic

ring (

http://planetmath.org/CyclicRing3

)

R

.

Proof.

Let

r

be a

generator

(

http://planetmath.org/Generator

) of the

additive group

of

R

. Then there exists

z

∈

ℤ

with

r

2

=

z

⁢

r

. If

z

≥

0

, then

z

is a behavior of

R

. Assume

z

<

0

. Note that

-

z

>

0

and

-

r

is also a generator of the additive group of

R

. Since

(

-

r

)

2

=

(

-

1

)

2

⁢

r

2

=

(

-

1

)

2

⁢

(

z

⁢

r

)

=

(

-

z

)

⁢

(

-

r

)

, it follows that

-

z

is a behavior of

R

. Thus, existence of behavior has been proven.

Let

a

and

b

be behaviors of

R

. Then there exist generators

s

and

t

of the additive group of

R

such that

s

2

=

a

⁢

s

and

t

2

=

b

⁢

t

. If

s

=

t

, then

a

⁢

s

=

s

2

=

t

2

=

b

⁢

t

=

b

⁢

s

, causing

a

=

b

. If

s

≠

t

, then it must be the case that

t

=

-

s

. (This follows from the fact that 1 and -1 are the only generators of

ℤ

.) Thus,

a

⁢

s

=

s

2

=

(

-

1

)

2

⁢

s

2

=

(

-

s

)

2

=

t

2

=

b

⁢

t

=

b

⁢

(

-

s

)

=

-

b

⁢

s

, causing

a

=

-

b

. Since

a

and

b

are nonnegative, it follows that

a

=

b

=

0

. Thus, uniqueness of behavior has been proven.
∎

Title

behavior exists uniquely (

infinite

case)

Canonical name

BehaviorExistsUniquelyinfiniteCase

Date of creation

2013-03-22 16:02:32

Last modified on

2013-03-22 16:02:32

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

11

Author

Wkbj79 (1863)

Entry type

Proof

Classification

msc 13A99

Classification

msc 16U99

Generated on Fri Feb 9 17:23:45 2018 by

LaTeXML
commutative ring

commutative ring

Let

(

X

,

+

,

⋅

)

be a ring. Since

(

X

,

+

)

is required to be an

abelian group

, the

operation

“

+

” necessarily is

commutative

.

This needs not to happen for “

⋅

”. Rings

R

where
“

⋅

” is commutative, that is,

x

⋅

y

=

y

⋅

x

for all

x

,

y

∈

R

, are called

commutative rings

.

The commutative rings are rings which are more like the fields
than other rings are, but there are certain dissimilarities. A
field has always a

multiplicative inverse

for each of its
nonzero

elements

, but the same needs not to be true for a
commutative ring. Further, in a commutative ring there may
exist

zero divisors

, i.e. nonzero elements having

product

zero.
Since the ideals of a commutative ring are
two-sided (

http://planetmath.org/Ideal

), the
these rings are more comfortable to handle than other rings.

The study of commutative rings is called

commutative algebra

.

Title

commutative ring

Canonical name

CommutativeRing

Date of creation

2013-11-12 18:26:58

Last modified on

2013-11-12 18:26:58

Owner

drini (3)

Last modified by

pahio (2872)

Numerical id

8

Author

drini (2872)

Entry type

Definition

Classification

msc 13A99

Related topic

GroupOfUnits

Related topic

ExampleOfRings

Generated on Fri Feb 9 17:23:49 2018 by

LaTeXML
criteria for cyclic rings to be isomorphic

criteria for cyclic rings to be isomorphic

Theorem

.

Two

cyclic rings

are

isomorphic

if and only if they have the same order and the same

behavior

.

Proof.

Let

R

be a cyclic ring with behavior

k

and

r

be a

generator

(

http://planetmath.org/Generator

) of the

additive group

of

R

with

r

2

=

k

⁢

r

. Also, let

S

be a cyclic ring.

If

R

and

S

have the same order and the same behavior, then let

s

be a generator of the additive group of

S

with

s

2

=

k

⁢

s

. Define

φ

:

R

→

S

by

φ

⁢

(

c

⁢

r

)

=

c

⁢

s

for every

c

∈

ℤ

. This map is clearly well defined and

surjective

. Since

R

and

S

have the same order,

φ

is

injective

. Since, for every

a

,

b

∈

ℤ

,

φ

⁢

(

a

⁢

r

)

+

φ

⁢

(

b

⁢

r

)

=

a

⁢

s

+

b

⁢

s

=

(

a

+

b

)

⁢

s

=

φ

⁢

(

(

a

+

b

)

⁢

r

)

=

φ

⁢

(

a

⁢

r

+

b

⁢

r

)

and

φ

⁢

(

a

⁢

r

)

⁢

φ

⁢

(

b

⁢

r

)

=

(

a

⁢

s

)

⁢

(

b

⁢

s

)

=

(

a

⁢

b

)

⁢

s

2

=

(

a

⁢

b

)

⁢

(

k

⁢

s

)

=

(

a

⁢

b

⁢

k

)

⁢

s

=

φ

⁢

(

(

a

⁢

b

⁢

k

)

⁢

r

)

=

φ

⁢

(

(

a

⁢

b

)

⁢

(

k

⁢

r

)

)

=

φ

⁢

(

(

a

⁢

b

)

⁢

r

2

)

=

φ

⁢

(

(

a

⁢

r

)

⁢

(

b

⁢

r

)

)

,

it follows that

φ

is an

isomorphism

.

Conversely

, let

ψ

:

R

→

S

be an isomorphism. Then

R

and

S

must have the same order. If

R

is

infinite

, then

S

is infinite, and

k

is a

nonnegative integer

. If

R

is finite, then

k

divides (

http://planetmath.org/

Divisibility

)

|

R

|

, which equals

|

S

|

. In either case,

k

is a candidate for the behavior of

S

. Since

r

is a generator of the additive group of

R

and

ψ

is an isomorphism,

ψ

⁢

(

r

)

is a generator of the additive group of

S

. Since

(

ψ

⁢

(

r

)

)

2

=

ψ

⁢

(

r

2

)

=

ψ

⁢

(

k

⁢

r

)

=

k

⁢

ψ

⁢

(

r

)

, it follows that

S

has behavior

k

.
∎

Title

criteria for cyclic rings to be isomorphic

Canonical name

CriteriaForCyclicRingsToBeIsomorphic

Date of creation

2013-03-22 16:02:39

Last modified on

2013-03-22 16:02:39

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

14

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 13A99

Classification

msc 16U99

Generated on Fri Feb 9 17:23:54 2018 by

LaTeXML