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However, an algebraic function of several variables may yield an algebraic number when applied to transcendental numbers if these numbers are not algebraically independent.

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## Some Related Sentences

However and algebraic

__However__

**,**most modern

__algebraic__geometry texts starting with Alexander Grothendieck's foundational EGA use the convention in this article .< ref > A notable exception

**to**modern

__algebraic__geometry texts following the conventions

**of**this article is Commutative algebra with a view toward

__algebraic__geometry / David Eisenbud ( 1995 ), which uses " h < sub > A </ sub >"

**to**mean the covariant hom-functor

**.**

__However__

**,**in the case

**of**

__algebraic__curves it is very common

**not**

**to**restrict the curve

**to**having points only defined over the real

**numbers**

**.**

__However__

**,**

**if**R is the ring

**of**

__algebraic__integers in

**an**

__algebraic__

**number**field

**,**or more generally a Dedekind domain

**,**the multiplication defined above turns the set

**of**fractional ideal classes into

**an**abelian group

**,**the ideal class group

**of**R

**.**The group property

**of**existence

**of**inverse elements follows easily from the fact that

**,**in a Dedekind domain

**,**every non-zero ideal ( except R ) is a product

**of**prime ideals

**.**

__However__

**,**the resulting

__algebraic__structure is

**not**a field

**,**and should

**not**be expected

**to**behave like one

**.**

__However__

**,**principal constructions

**of**such more detailed theories for small

__algebraic__

**number**fields

**are**

**not**extendable

**to**the general case

**of**

__algebraic__

**number**fields

**,**and different conceptual principles

**are**in use in the general class field theory

**.**

If the base field F is the field R

**of**real**numbers****,**then x < sup > 2 </ sup > + y < sup > 2 </ sup > = − 1 defines**an**__algebraic__extension field**of**R ( x ), but the corresponding curve considered as a locus has no points in R**.**__However__**,**it does have points defined over the__algebraic__closure C**of**R**.**__However__

**,**the user also has the option

**of**typing in moves in

__algebraic__notation

**,**ideal for laptop users with only the built in touchpad as their mouse

**.**

__However__

**,**in contrast

**to**the situation common for isomorphisms in

**an**

__algebraic__setting

**,**the composition

**of**the functor and its " inverse " is

**not**necessarily the identity mapping

**.**

__However__

**,**a purely

__algebraic__theory was first developed by

**,**with Armand Borel as one

**of**its pioneers

**.**

__However__

**,**for univariate

__algebraic__equations

**,**the

**number**

**of**solutions is finite and all solutions

**,**

**are**contained in any

**algebraically**closed field containing the coefficients

**,**for example

**,**the field

**of**complex

**numbers**in case

**of**equations over the rationals

**.**

and e would therefore be

__algebraic__**of**degree p**.**__However__e is**transcendental****,**so Ω must be irrational**.**__However__

**,**the converse is true

**if**X and Y contain only

__algebraic__

**numbers**and their size is at least 2 × 2

**.**

__However__it can also be used

**to**suggest particular test-statistics that might be

**of**interest or

**to**suggest simplified tests — for this one considers

__algebraic__manipulation

**of**the ratio

**to**see

**if**there

**are**key statistics in it related

**to**the size

**of**the ratio ( i

**.**e

**.**whether a large statistic corresponds

**to**a small ratio or

**to**a large one ).

__However__

**,**the Garden

**of**Eden theorem can be generalized beyond Euclidean spaces

**,**

**to**cellular automata defined on the elements

**of**

**an**amenable group or a sofic group ; the proof

**of**this generalization uses the Ax – Grothendieck theorem

**,**

**an**analogous relation between injectivity and bijectivity in

__algebraic__geometry

**.**

__However__

**,**the

__algebraic__properties

**of**this structure

**are**somewhat different from those

**of**classical propositional logic in that the principle

**of**distributivity fails

**.**

__However__

**,**under certain conditions

__algebraic__solutions can be obtained ; for example

**,**the equation can be solved as

__However__

**,**the word fraction is also used

**to**describe mathematical expressions that

**are**

**not**rational

**numbers**

**,**for example

__algebraic__fractions ( quotients

**of**

__algebraic__expressions ), and expressions that contain irrational

**numbers**

**,**such as √ 2 / 2 ( see square root

**of**2 ) and π / 4 ( see proof that π is irrational ).

However and function

"

__However__**,**as Stocking notes**,**Tylor mainly concerned himself with describing and mapping the distribution**of**particular elements**of**culture**,**rather than with the larger__function__**,**and he generally seemed**to**assume a Victorian idea**of**progress rather than the idea**of**non-directional**,**multilineal cultural development proposed by later anthropologists**.**__However__

**,**in CL it is necessary

**to**explicitly refer

**to**the

__function__namespace

**when**passing a

__function__as

**an**argument — which is also a common occurrence

**,**as in the example above

**.**

__However__

**,**alternative analysis suggests that the nominal particles do

**not**

__function__as clitics

**,**but as phrasal affixes

**.**

__However__

**,**it is known that cortisone impairs memory

__function__by reducing the blood flow in the right parahippocampal gyrus

**,**left visual cortex

**,**and the Cerebellum

**.**

__However__

**,**

**if**there is no " obvious " Euclidean

__function__

**,**then determining whether R is a PID is generally a much easier problem than determining whether it is a Euclidean domain

**.**

__However__

**,**it took a few more decades until the new regulation was universally accepted and the new court actually began

**to**

__function__; only in 1512 would the Imperial Circles be finalised

**.**

__However__

**,**they did

**not**make any conclusions regarding the specific

__function__

**of**DNA as hereditary material

**,**and only said that it must have some undefined role

**.**

__However__

**,**the expression sin < sup >− 1 </ sup > x usually does

**not**represent the multiplicative inverse

**to**sin x

**,**but the inverse

**of**the sine

__function__

**applied**

**to**x ( actually a partial inverse ; see below ).

__However__in Stirner's political thinking there can be found a surprising convergence with poststructuralist theory

**,**particularly with regard

**to**the

__function__

**of**power

**.**

__However__

**,**the latter chip was necessary in order

**to**provide the FERR signal

**to**the mainboard and appear

**to**

__function__as a normal floating point unit

**.**

__However__

**,**the reason for this correction

**to**the partition

__function__remained obscure until the discovery

**of**quantum mechanics

**.**

__However__

**,**in the untyped lambda calculus

**,**there's no way

**to**prevent our

__function__from being

**applied**

**to**truth values

**,**or strings

**,**for instance

**.**

__However__

**,**its

__function__has evolved from and replaced the macro languages that were originally included in some

**of**

**these**applications

**.**

__However__

**,**on average and across the most commonly studied

**variables**

**,**acquiring firms ' financial performance does

**not**positively change as a

__function__

**of**their acquisition activity

**.**

__However__

**,**like methadone

**,**buprenorphine produces similar cognitive dehabilitation in multiple areas

**of**mental

__function__in both memory and timed choice task tests

**,**which

**may**persist after cessation

**of**substitution treatment

**.**

__However__

**,**none

**of**this research and criticism was originally known

**to**most

**of**the English-language commentators

**,**by

__function__

**of**the dates

**when**they were writing and

**,**

**to**some extent

**,**

**of**the language in which it was written

**.**

__However__

**,**in the context

**of**topology

**,**sequences do

**not**fully encode all information about a

__function__between topological spaces

**.**

__However__

**,**in this circuit V < sub >−</ sub > is a

__function__

**of**V < sub > out </ sub > because

**of**the negative feedback through the R < sub > 1 </ sub > R < sub > 2 </ sub > network

**.**

__However__licensing was gradually liberalised after the 1960s

**,**until contested licensing applications became very rare

**,**and the remaining administrative

__function__was transferred

**to**Local Authorities in 2005

**.**

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