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Every field and every polynomial ring over a field ( in arbitrarily many variables ) is a reduced ring.

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## Some Related Sentences

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__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 every

__Every__legislator from Brasstown Bald to Folkston

**is**going to have his

__every__vote subjected to the closest scrutiny as

**a**test of his political allegiances, not his convictions

**.**

__Every__pitch

**in**the game brought forth

**a**howl from the enraptured audience

**and**

__every__fly ball the visitors dropped

**(**

**and**because their right fielder was still

**a**little fuzzy from drink, they dropped

**many**

**)**called forth yelps of derision

**.**

** 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**.**__Every__natural-born citizen of

**a**foreign state who

**is**also an American citizen

**and**

__every__natural-born American citizen who

**is**

**a**citizen of

**a**foreign land owes

**a**double allegiance, one to the United States,

**and**one to his homeland

**(**

**in**the event of an immigrant becoming

**a**citizen of the US ), or to his adopted land

**(**

**in**the event of an emigrant natural born citizen of the US becoming

**a**citizen of another nation ).

*

__Every__quadratic Bézier curve**is**also**a**cubic Bézier curve,**and**more generally,__every__degree n Bézier curve**is**also**a**degree m curve for any m > n**.**In detail,**a**degree n curve with control points P < sub > 0 </ sub >, …, P < sub > n </ sub >**is**equivalent**(**including the parametrization**)**to the degree n + 1 curve with control points P '< sub > 0 </ sub >, …, P '< sub > n + 1 </ sub >, where**.**
*

__Every__unital real Banach algebra with no zero divisors,**and****in**which__every__principal ideal**is**closed,**is**isomorphic to the reals, the complexes, or the quaternions**.**__Every__cleric

**in**Holy Orders

**and**

__every__member of

**a**religious order must publicly join

**in**or privately read aloud

**(**i

**.**e

**.**using the lips as well as the eyes — it takes about two hours

**in**this way

**)**the whole of the Breviary services allotted for each day

**.**

__Every__regular language

**is**context-free,

__every__context-free language, not containing the empty string,

**is**context-sensitive

**and**

__every__context-sensitive language

**is**recursive

**and**

__every__recursive language

**is**recursively enumerable

**.**

__Every__grammar

**in**Chomsky normal form

**is**context-free,

**and**conversely,

__every__context-free grammar can be transformed into an equivalent one which

**is**

**in**Chomsky normal form

**.**

__Every__corporation, whether financial or union, as well as

__every__division of the administration, were set up as branches of the party, the CEOs, Union leaders,

**and**division directors being sworn-in as section presidents of the party

**.**

__Every__gun that

**is**made,

__every__warship launched,

__every__rocket fired signifies,

**in**the final sense,

**a**theft from those who hunger

**and**are not fed, those who are cold

**and**are not clothed

**.**

__Every__mental phenomenon,

__every__psychological act, has

**a**content,

**is**directed at an object

**(**the intentional object ).

" More immediately, though, he called for

**a**massive build-up of U**.**S**.**arms production: "__Every__realist knows that the democratic way of life**is**at this moment being ' directly assailed**in**__every__part of the world … The need of the moment**is**that our actions**and**our policy should be devoted primarily — almost exclusively — to meeting this foreign peril**.**__Every__continuous map f: X → Y induces an algebra homomorphism C

**(**f

**)**: C

**(**Y

**)**→ C

**(**X

**)**by the rule C

**(**f )( φ

**)**= φ o f for

__every__φ

**in**C

**(**Y ).

__Every__galaxy of sufficient mass

**in**the Local Group has an associated group of globular clusters,

**and**almost

__every__large galaxy surveyed has been found to possess

**a**system of globular clusters

**.**

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