[permalink] [id link]

Every ordered field contains an ordered subfield that is isomorphic to the rational numbers.

from
Wikipedia

## Some Related Sentences

Every and ordered

** Zorn's lemma:

__Every__non-empty partially__ordered__set in which every chain ( i**.**e**.**totally__ordered__subset ) has**an**upper bound**contains**at least one maximal element**.**
The restricted principle "

__Every__partially__ordered__set has a maximal totally__ordered__subset "**is**also equivalent**to**AC over ZF**.**__Every__

__ordered__

**field**

**is**a formally real

**field**, i

**.**e., 0 cannot be written as a sum of nonzero squares

**.**

*

__Every__non-empty set of left ideals of R, partially__ordered__by inclusion, has a maximal element with respect**to**set inclusion**.**
*

__Every__totally__ordered__set**that****is**a bounded lattice**is**also a Heyting algebra, where**is**equal**to**when, and 1 otherwise**.**__Every__time she entered, song typical of

**the**Brazilian northeast would play, stopping only when

__ordered__by Cirene herself

**.**

Every and field

__Every__

__field__has

**an**algebraic extension which

**is**algebraically closed ( called its algebraic closure ), but proving this in general requires some form of

**the**axiom of choice

**.**

__Every__time

**an**MTA receives

**an**email message, it adds a < tt > Received </ tt > trace header

__field__

**to**

**the**top of

**the**header of

**the**message, thereby building a sequential record of MTAs handling

**the**message

**.**

__Every__instruction consists of a 9-bit opcode, a 4-bit register code, and a 23-bit effective address

__field__, which consists in turn of a 1-bit indirect bit, a 4-bit register code, and

**an**18-bit offset

**.**

__Every__

__field__theory of particle physics

**is**based on certain symmetries of nature whose existence

**is**deduced from observations

**.**

__Every__planetary body ( including

**the**Earth )

**is**surrounded by its own gravitational

__field__, which exerts

**an**attractive force on all objects

**.**

__Every__prime ideal P in a Boolean ring R

**is**maximal:

**the**quotient ring R / P

**is**

**an**integral domain and also a Boolean ring, so it

**is**

**isomorphic**

**to**

**the**

__field__F < sub > 2 </ sub >, which shows

**the**maximality of P

**.**Since maximal ideals are always prime, prime ideals and maximal ideals coincide in Boolean rings

**.**

__Every__year,

**the**British Political Studies Association awards

**the**Walter Bagehot Prize for

**the**best dissertation in

**the**

__field__of government and public administration

**.**

__Every__polynomial in can be factorized into polynomials

**that**are irreducible over F

**.**This factorization

**is**unique up

**to**permutation of

**the**factors and

**the**multiplication of

**the**factors by nonzero constants from F ( because

**the**ring of polynomials over a

__field__

**is**a unique factorization domain whose units are

**the**nonzero constant polynomials ).

__Every__quadratic form q in n variables over a

__field__of characteristic not equal

**to**2

**is**equivalent

**to**a diagonal form

__Every__kind of material has unique magnetic properties, even those

**that**we do not think of as being “ magnetic .” Different materials below

**the**ground can cause local disturbances in

**the**Earth ’ s magnetic

__field__

**that**are detectable with sensitive magnetometers

**.**

Every and contains

__Every__computer

__contains__

**an**internal clock

**that**regulates

**the**rate at which instructions are executed and synchronizes all

**the**various computer components

**.**

Cantor points out

**that**his constructions prove more — namely, they provide a new proof of Liouville's theorem:__Every__interval__contains__infinitely many transcendental**numbers****.**
The revised edition of Robyn was released in

**the**UK in April 2007, and__contains__two new tracks —" With__Every__Heartbeat " ( a collaboration with Kleerup ) and " Cobrastyle " ( a cover of a 2006 single by Swedish rockers Teddybears )— alongside slightly altered versions of two of**the**original songs**.**
#

__Every__simple path from a given node**to**any of its descendant leaves__contains__**the**same number of black nodes**.**__Every__synset

__contains__a group of synonymous words or collocations ( a collocation

**is**a sequence of words

**that**go together

**to**form a specific meaning, such as " car pool "); different senses of a word are in different synsets

**.**

*

__Every__connected graph G admits a spanning tree, which**is**a tree**that**__contains__every vertex of G and whose edges are edges of G**.**__Every__one has higher glucose levels for one or two hours after eating food which

__contains__some types of carbohydrates

**.**

__Every__temple of The Church of Jesus Christ of Latter-day Saints ( LDS Church )

__contains__a baptismal font on twelve oxen

**that**

**is**modeled after

**the**molten sea

**.**

__Every__issue of GAMES Magazine

__contains__a large crossword with a double clue list, under

**the**title The World's Most Ornery Crossword ; both lists are straight and arrive at

**the**same solution, but one list

**is**significantly more challenging than

**the**other

**.**

__Every__gap sequence

**that**

__contains__1 yields a correct sort ; however,

**the**properties of thus obtained versions of Shellsort may be very different

**.**

The offset part of

**the**logical address__contains__**an**offset inside**the**segment, i**.**e**.****the**physical address can be calculated as ( if**the**address line A20**is**enabled ), respectively ( segment_part × 16 + offset ) mod 2 < sup > 20 </ sup > ( if A20**is**off )__Every__segment has a size of 2 < sup > 16 </ sup > bytes**.**__Every__chapter

__contains__a comprehensive compilation all of

**the**published examples of

**the**reaction organized in tables according

**to**

**the**structure of

**the**starting material

**.**

__Every__Riemann surface

**is**a two-dimensional real analytic manifold ( i

**.**e., a surface ), but it

__contains__more structure ( specifically a complex structure ) which

**is**needed for

**the**unambiguous definition of holomorphic functions

**.**

0.389 seconds.