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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 ).

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## Some Related Sentences

Every and polynomial

__Every__output

**of**an encoder

**can**

**be**described

**by**its own transfer function, which

**is**closely related

**to**

**the**generator

__polynomial__

**.**

*

__Every__root**of****a**monic__polynomial__**whose**coefficients**are**algebraic integers**is**itself an algebraic integer**.**__Every__delta operator ' has

**a**

**unique**sequence

**of**" basic

**polynomials**",

**a**

__polynomial__sequence defined

**by**three conditions:

*

__Every__**irreducible**closed subset**of**P < sup > n </ sup >( k )**of**codimension one**is****a**hypersurface ; i**.**e.,**the**zero set**of**some homogeneous__polynomial__**.**
*

__Every__Jacobi-like__polynomial__sequence**can**have its**domain**shifted**and**/ or scaled so**that**its interval**of**orthogonality**is**,**and**has Q
*

__Every__Laguerre-like__polynomial__sequence**can**have its**domain**shifted, scaled,**and**/ or reflected so**that**its interval**of**orthogonality**is**,**and**has Q =
*

__Every__Hermite-like__polynomial__sequence**can**have its**domain**shifted**and**/ or scaled so**that**its interval**of**orthogonality**is**,**and**has Q__Every__

**field**

**and**every

__polynomial__

**ring**

**over**

**a**

**field**

**(**

**in**arbitrarily many variables )

**is**

**a**reduced

**ring**

**.**

In mathematics, an integer-valued

__polynomial__**(**also known as**a**numerical__polynomial__) P**(**t )**is****a**__polynomial__**whose**value P**(**n )**is**an integer for every integer n**.**__Every____polynomial__with integer coefficients**is**integer-valued, but**the**converse**is**not true**.**

Every and can

__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

**.**

__Every__information exchange between living organisms — i

**.**e

**.**transmission

**of**signals

**that**involve

**a**living sender

**and**receiver

__can__

**be**considered

**a**form

**of**communication ;

**and**even primitive creatures such as corals

**are**competent

**to**communicate

**.**

__Every__context-sensitive grammar which does not generate

**the**empty string

__can__

**be**transformed

**into**an equivalent one

**in**Kuroda normal form

**.**

__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__module

**over**

**a**division

**ring**has

**a**basis ; linear maps between finite-dimensional modules

**over**

**a**division

**ring**

__can__

**be**described

**by**matrices,

**and**

**the**Gaussian elimination algorithm remains applicable

**.**

__Every__entire function

__can__

**be**represented as

**a**power series

**that**converges uniformly on compact sets

**.**

Group actions / representations:

__Every__group G__can__**be**considered as**a**category with**a**single object**whose**morphisms**are****the**elements**of**G**.**A functor**from**G**to**Set**is**then nothing but**a**group action**of**G on**a**particular set, i**.**e**.****a**G-set**.**__Every__sequence

__can__, thus,

**be**read

**in**three reading frames, each

**of**which will produce

**a**different amino acid sequence

**(**

**in**

**the**given example, Gly-Lys-Pro, Gly-Asn, or Glu-Thr, respectively

**).**

__Every__hyperbola

**is**congruent

**to**

**the**origin-centered East-West opening hyperbola sharing its same eccentricity ε

**(**its shape, or degree

**of**" spread "),

**and**

**is**also congruent

**to**

**the**origin-centered North-South opening hyperbola with identical eccentricity ε —

**that**

**is**, it

__can__

**be**rotated so

**that**it opens

**in**

**the**desired direction

**and**

__can__

**be**translated

**(**rigidly moved

**in**

**the**plane ) so

**that**it

**is**centered at

**the**origin

**.**

__Every__holomorphic function

__can__

**be**separated

**into**its real

**and**imaginary parts,

**and**each

**of**these

**is**

**a**solution

**of**Laplace's equation on R < sup > 2 </ sup >.

__Every__species

__can__

**be**given

**a**

**unique**

**(**

**and**, one hopes, stable ) name, as compared with common names

**that**

**are**often neither

**unique**nor consistent

**from**place

**to**place

**and**language

**to**language

**.**

__Every__vector v

**in**determines

**a**linear map

**from**R

**to**taking 1

**to**v, which

__can__

**be**thought

**of**as

**a**Lie algebra homomorphism

**.**

__Every__use

**of**modus tollens

__can__

**be**converted

**to**

**a**use

**of**modus ponens

**and**one use

**of**transposition

**to**

**the**premise which

**is**

**a**material implication

**.**

__Every__adult, healthy, sane Muslim who has

**the**financial

**and**physical capacity

**to**travel

**to**Mecca

**and**

__can__make arrangements for

**the**care

**of**his / her dependants during

**the**trip, must perform

**the**Hajj once

**in**

**a**lifetime

**.**

*

__Every__finite topological space gives rise**to****a**preorder on its points,**in**which x ≤ y if**and**only if x belongs**to**every neighborhood**of**y,**and**every finite preorder__can__**be**formed as**the**specialization preorder**of****a**topological space**in**this way**.**
*

__Every__preorder__can__**be**given**a**topology,**the**Alexandrov topology ;**and**indeed, every preorder on**a**set**is****in**one-to-one correspondence with an Alexandrov topology on**that**set**.**__Every__binary relation R on

**a**set S

__can__

**be**extended

**to**

**a**preorder on S

**by**taking

**the**transitive closure

**and**reflexive closure, R < sup >+=</ sup >.

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