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If K is a number field, its ring of integers is the subring of algebraic integers in K, and is frequently denoted as O < sub > K </ sub >.

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## Some Related Sentences

If and K

__If__

**a**description

**of**s

**,**d ( s ),

**is**

**of**minimal length ( i. e. it uses

**the**fewest

**number**

**of**characters ), it

**is**called

**a**minimal description

**of**s. Thus

**,**

**the**length

**of**d ( s ) ( i. e.

**the**

**number**

**of**characters

**in**

**the**description )

**is**

**the**Kolmogorov complexity

**of**s

**,**written

__K__( s ).

Theorem:

__If____K__**<****sub****>**1**</****sub****>****and**__K__**<****sub****>**2**</****sub****>**are**the**complexity functions relative to description languages L**<****sub****>**1**</****sub****>****and**L**<****sub****>**2**</****sub**>, then there**is****a**constant c – which depends only on**the**languages L**<****sub****>**1**</****sub****>****and**L**<****sub****>**2**</****sub****>**chosen – such that__If__

**a**

**is**

**algebraic**over

__K__

**,**then

__K__

**,**

**the**set

**of**all polynomials

**in**

**a**with coefficients

**in**

__K__

**,**

**is**not only

**a**

**ring**but

**a**

**field**: an

**algebraic**extension

**of**

__K__which has finite degree over

__K__. In

**the**special case where

__K__= Q

**is**

**the**

**field**

**of**rational numbers

**,**Q

**is**an example

**of**an

**algebraic**

**number**

**field**.

__If__X

**and**Y are Banach spaces over

**the**same ground

**field**

__K__

**,**

**the**set

**of**all continuous K-linear maps T: X → Y

**is**

**denoted**by B ( X

**,**Y ).

__If__X

**is**

**a**Banach space

**and**

__K__

**is**

**the**underlying

**field**( either

**the**real or

**the**complex numbers ), then

__K__

**is**itself

**a**Banach space ( using

**the**absolute value

**as**norm )

**and**we can define

**the**continuous dual space

**as**X ′ = B ( X

**,**

__K__),

**the**space

**of**continuous linear maps into

__K__.

__If__their existence past

**the**

__K__– T boundary can be confirmed

**,**these hadrosaurids would be considered

**a**Dead Clade Walking.

Likewise

**,****a**functor from G to**the**category**of**vector spaces**,**Vect**<****sub****>**__K__**</****sub**>,**is****a**linear representation**of**G. In general**,****a**functor G → C can be considered**as**an " action "**of**G on an object**in****the**category C.__If__C**is****a**group**,**then this action**is****a**group homomorphism.__If__

__K__

**is**

**a**subset

**of**ker ( f ) then there exists

**a**unique homomorphism h: G /

__K__→ H such that f = h φ.

__If__G = GL

**<**

**sub**>*</

**sub**>(

__K__), then

**the**set

**of**natural numbers

**is**

**a**proper subset

**of**G

**<**

**sub**

**>**0

**</**

**sub**>, since for each natural

**number**n

**,**there

**is**

**a**corresponding identity matrix

**of**dimension n. G ( m

**,**n )

**is**empty unless m = n

**,**

**in**which case it

**is**

**the**set

**of**all nxn invertible matrices.

__If__

**the**X ion

**is**

__K__or Na

**,**

**the**mica

**is**

**a**" common " mica

**,**whereas if

**the**X ion

**is**Ca

**,**

**the**mica

**is**classed

**as**

**a**" brittle " mica.

:

__If__**the**temperature T**of****a**body**is**quantified**as**300 kelvin (**in**which T**is****the**quantity symbol**,**300**the**value**,****and**__K__**is****the**unit ), this**is**written

If and is

__If__his dancers are sometimes made to look

**as**if they might be creatures from Mars

**,**this

__is__consistent with his intention

**of**placing them

**in**

**the**orbit

**of**another world

**,**

**a**world

**in**which they are freed

**of**their pedestrian identities.

__If__

**a**work

__is__divided into several large segments

**,**

**a**last-minute drawing

**of**random numbers may determine

**the**order

**of**

**the**segments for any particular performance.

__If__Wilhelm Reich

__is__

**the**Moses who has led them out

**of**

**the**Egypt

**of**sexual slavery

**,**Dylan Thomas

__is__

**the**poet who offers them

**the**Dionysian dialectic

**of**justification for their indulgence

**in**liquor

**,**marijuana

**,**sex

**,**

**and**jazz.

__If__he

__is__

**the**child

**of**nothingness

**,**if he

__is__

**the**predestined victim

**of**an age

**of**atomic wars

**,**then he will consult only his own organic needs

**and**go beyond good

**and**evil.

__If__he thus achieves

**a**lyrical

**,**dreamlike

**,**drugged intensity

**,**he pays

**the**price for his indulgence by producing work -- Allen Ginsberg's `` Howl ''

__is__

**a**striking example

**of**this tendency -- that

__is__disoriented

**,**Dionysian but without depth

**and**without Apollonian control.

__If__love reflects

**the**nature

**of**man

**,**

**as**Ortega Y Gasset believes

**,**if

**the**person

**in**love betrays decisively what he

__is__by his behavior

**in**love

**,**then

**the**writers

**of**

**the**beat generation are creating

**a**new literary genre.

__If__

**the**existent form

__is__to be retained new factors that reinforce it must be introduced into

**the**situation.

__If__we remove ourselves for

**a**moment from our time

**and**our infatuation with mental disease

**,**isn't there something absurd about

**a**hero

**in**

**a**novel who

__is__defeated by his infantile neurosis??

__If__many

**of**

**the**characters

**in**contemporary novels appear to be

**the**bloodless relations

**of**characters

**in**

**a**case history it

__is__because

**the**novelist

__is__often forgetful today that those things that we call character manifest themselves

**in**surface behavior

**,**that

**the**ego

__is__still

**the**executive agency

**of**personality

**,**

**and**that all we know

**of**personality must be discerned through

**the**ego.

__If__our sincerity

__is__granted

**,**

**and**it

__is__granted

**,**

**the**discrepancy can only be explained by

**the**fact that we have come to believe hearsay

**and**legend about ourselves

**in**preference to an understanding gained by earnest self-examination.

If and number

__If__

**in**any one calculation Ptolemy had had to invoke 83 epicycles all at once

**,**while Copernicus never required more than one third this

__number__

**,**then (

**in**

**the**sense obvious to Margenau ) Ptolemaic astronomy would be simpler than Copernican.

__If__your state has no provisions for

**the**numbering

**of**pleasure boats

**,**you must apply for

**a**

__number__from

**the**U.S. Coast Guard for any kind

**of**boat with mechanical propulsion rated at more than 10 horsepower before it can be used on Federal waterways.

__If__

**a**nation wished to get

**a**head start

**in**physical fitness over all other nations

**,**it would start

**its**kindergarten students on

**a**program

**of**gymnastics

**the**day they entered

**and**thus eliminate

**a**large

__number__

**of**

**the**problems that plague American schools.

__If__no specific organization plan exists limiting

**the**

__number__

**of**scientists at each salary level

**,**

**the**result

**is**

**a**department top-heavy with high-level

**,**high-salaried personnel ''.

__If__we thus spent our very first day

**in**

**the**midst

**of**

**a**large

__number__

**of**your people honoring

**a**new hero

**and**

**a**great national achievement

**,**our last day

**,**to us at least

**,**was equally impressive

**and**very moving

**,**even though

**the**crowds were absent

**and**there was almost complete silence.

__If__

**a**wedge-shaped coating

**of**increasing thickness

**is**removed from

**a**substrate by an instrument like

**the**Hesiometer with

**a**knife

**of**constant rake angle

**,**

**a**

__number__

**of**removal mechanisms are often observed which depend upon

**the**thickness

**of**

**the**coating.

__If__all

**the**operating variables were varied simultaneously

**,**Af operations would be required to do

**the**same job

**,**

**and**

**as**R increases this increases very much more rapidly than

**the**

__number__

**of**operations required by

**the**dynamic program.

__If__one takes

**the**middle

__number__

**,**5

**,**

**and**multiplies it by 3 (

**the**base

__number__

**of**

**the**magic square

**of**three )

**,**

**the**result

**is**15

**,**which

**is**also

**the**constant sum

**of**all

**the**rows

**,**columns

**,**

**and**two main diagonals.

__If__so

**,**it might be worth while to assign

**a**future jazz show to

**a**different department -- one with enough confidence

**in**

**the**musical material to cut down on

**the**

__number__

**of**performers

**and**give them

**a**little room to display their talents.

*

__If__it**is**required to use**a**single__number__X**as**an estimate for**the**value**of**numbers**,**then**the**arithmetic mean does this best**,****in****the**sense**of**minimizing**the**sum**of**squares ( x**<****sub****>**i**</****sub****>**− X )< sup**>**2**</**sup**>****of****the**residuals.
* Let Q be

**a**set enclosed between two step regions S**and**T. A step region**is**formed from**a**finite union**of**adjacent rectangles resting on**a**common base**,**i. e. S ⊆ Q ⊆ T.__If__there**is****a**unique__number__c such that**a**( S ) ≤ c ≤**a**( T ) for all such step regions S**and**T**,**then**a**( Q )__If__

**a**) testimonies conflict one another

**,**b ) there are

**a**small

__number__

**of**witnesses

**,**c )

**the**speaker has no integrity

**,**d )

**the**speaker

**is**overly hesitant or bold

**,**or e )

**the**speaker

**is**known to have motives for lying

**,**then

**the**epistemologist has reason to be skeptical

**of**

**the**speaker's claims.

__If__we take

**in**our hand any volume ;

**of**divinity or school metaphysics

**,**for instance ; let us ask

**,**Does it contain any abstract reasoning concerning quantity or

__number__?

__If__

**the**two carbon atoms at

**the**double bond are linked to

**a**different

__number__

**of**hydrogen atoms

**,**

**the**halogen

**is**found preferentially at

**the**carbon with fewer hydrogen substituents ( Markovnikov's rule ).

__If__F

**is**an antiderivative

**of**f

**,**

**and**

**the**function f

**is**defined on some interval

**,**then every other antiderivative G

**of**f differs from F by

**a**constant: there exists

**a**

__number__C such that G ( x ) = F ( x ) + C for all x.

__If__

**the**game has been scored correctly

**,**

**the**total

__number__

**of**plate appearances for

**a**team should equal

**the**total

**of**that team's runs

**,**men left on base

**,**

**and**men put out.

__If__

**the**other player also knows to offer sacrifices

**,**

**the**expert also has to manipulate

**the**

__number__

**of**available sacrifices through earlier play.

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