[permalink] [id link]

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.

from
Wikipedia

## Some Related Sentences

Every and field

__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__ordered

__field__

**is**a formally real

__field__, i

**.**e., 0 cannot be written as a sum

**of**nonzero squares

**.**

__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 has

__Every__woman

__has__had

**the**experience

**of**saying no when she meant yes, and saying yes when she meant no

**.**

__Every__detail

**in**his interpretation

__has__been beautifully thought out, and

**of**these I would especially cite

**the**delicious laendler touch

**the**pianist brings to

**the**fifth variation

**(**

**an**obvious indication that he

**is**playing with Viennese musicians ), and

**the**gossamer shading throughout

**.**

__Every__family

**of**Riviera Presbyterian Church

__has__been asked to read

**the**Bible and pray together daily during National Christian Family Week and to undertake one project

**in**

**which**all members

**of**

**the**family participate

**.**

__Every__community, if it

**is**alive

__has__a spirit, and that spirit

**is**

**the**center

**of**

**its**unity and identity

**.**

__Every__such subset

__has__a smallest element, so to specify our

**choice**function we can simply say that it maps each set to

**the**least element

**of**that set

**.**

** 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**.**
** Tukey's lemma:

__Every__non-empty collection**of**finite character__has__a maximal element with respect to inclusion**.**
*

__Every__continuous functor on a small-complete category**which**satisfies**the**appropriate solution set condition__has__a left-adjoint**(****the**Freyd adjoint functor theorem ).__Every__unit

**of**length

__has__a corresponding unit

**of**area, namely

**the**area

**of**a square with

**the**given side length

**.**

__Every__ATM cell

__has__

**an**8-or 12-bit Virtual Path Identifier

**(**VPI ) and 16-bit Virtual Channel Identifier

**(**VCI ) pair defined

**in**

**its**header

**.**

0.110 seconds.