Absolute value and Algebraically closed field: Difference between pages

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In [[abstract algebra]], an '''algebraically closed field''' ''F'' contains a [[Zero of a function|root]] for every [[Degree of a polynomial|non-constant polynomial]] in ''F''[''x''], the [[ring of polynomials]] in the variable ''x'' with coefficients in ''F''.
[[Image:Khoang cach tren duong thang thuc.png|thumb|The absolute value of a number may be thought of as its distance from zero.]]
In [[mathematics]], the '''absolute value''' (or '''modulus''') {{math|{{!}}''x''{{!}}}} of a [[real number]] {{mvar|x}} is the [[non-negative]] value of {{mvar|x}} without regard to its [[sign (mathematics)|sign]]. Namely, {{math|1={{!}}''x''{{!}} = ''x''}} for a [[positive number|positive]] {{mvar|x}}, {{math|1={{!}}''x''{{!}} = [[additive inverse|−''x'']]}} for a [[negative number|negative]] {{mvar|x}}, and {{math|1={{!}}0{{!}} = 0}}. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its [[distance]] from zero.


Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example an absolute value is also defined for the [[complex number]]s, the [[quaternion]]s, [[ordered ring]]s, [[Field (mathematics)|fields]] and [[vector space]]s. The absolute value is closely related to the notions of [[magnitude (mathematics)|magnitude]], [[distance]], and [[Norm (mathematics)|norm]] in various mathematical and physical contexts.
==Examples==
As an example, the [[field (mathematics)|field]] of [[real number]]s is not algebraically closed, because the polynomial equation ''x''<sup>2</sup>&nbsp;+&nbsp;1&nbsp;=&nbsp;0&nbsp; has no solution in real numbers, even though all its coefficients (1 and 0) are real.  The same argument proves that no subfield of the real field is algebraically closed; in particular, the field of [[rational number]]s is not algebraically closed. Also, no [[finite field]] ''F'' is algebraically closed, because if ''a''<sub>1</sub>, ''a''<sub>2</sub>, …, ''a<sub>n</sub>'' are the elements of ''F'', then the polynomial (''x''&nbsp;&minus;&nbsp;''a''<sub>1</sub>)(''x''&nbsp;&minus;&nbsp;''a''<sub>2</sub>)&nbsp;···&nbsp;(''x''&nbsp;&minus;&nbsp;''a''<sub>''n''</sub>)&nbsp;+&nbsp;1
has no zero in ''F''. By contrast, the [[fundamental theorem of algebra]] states that the field of [[complex number]]s is algebraically closed. Another example of an algebraically closed field is the field of (complex) [[algebraic number]]s.


==Terminology and notation==
==Equivalent properties==
[[Jean-Robert Argand]] introduced the term "module", meaning 'unit of measure' in French, in 1806 specifically for the ''complex'' absolute value<ref name=oed>[[Oxford English Dictionary]], Draft Revision, June 2008</ref><ref>[http://www.amazon.com/gp/reader/0691027951 Nahin], [http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Argand.html O'Connor and Robertson], and [http://functions.wolfram.com/ComplexComponents/Abs/35/ functions.Wolfram.com.]; for the French sense, see [[Dictionnaire de la langue française (Littré)|Littré]], 1877</ref> and it was borrowed into English in 1866 as the Latin equivalent "modulus".<ref name=oed /> The term "absolute value" has been used in this sense since at least 1806 in French<ref>[[Lazare Nicolas Marguerite Carnot|Lazare Nicolas M. Carnot]], ''Mémoire sur la relation qui existe entre les distances respectives de cinq point quelconques pris dans l'espace'', p.&nbsp;105 [http://books.google.com/books?id=YyIOAAAAQAAJ&pg=PA105 at Google Books]</ref> and 1857 in English.<ref>James Mill Peirce, ''A Text-book of Analytic Geometry'' [http://books.google.com/books?id=RJALAAAAYAAJ&pg=PA42 at Google Books].  The oldest citation in the 2nd edition of the Oxford English Dictionary is from 1907. The term "absolute value" is also used in contrast to "relative value".</ref> The notation {{math|{{!}}''x''{{!}}}} was introduced by [[Karl Weierstrass]] in 1841.<ref>Nicholas J. Higham, ''Handbook of writing for the mathematical sciences'', SIAM. ISBN 0-89871-420-6, p.&nbsp;25</ref> Other names for ''absolute value'' include "the numerical value"<ref name=oed /> and "the magnitude".<ref name=oed />
Given a field ''F'', the assertion “''F'' is algebraically closed” is equivalent to other assertions:


The same notation is used with sets to denote [[cardinality]]; the meaning depends on context.
===The only irreducible polynomials are those of degree one===
The field ''F'' is algebraically closed if and only if the only [[irreducible polynomial]]s in the [[polynomial ring]] ''F''[''x''] are those of degree one.


==Definition and properties==
The assertion “the polynomials of degree one are irreducible” is trivially true for any field. If ''F'' is algebraically closed and ''p''(''x'') is an irreducible polynomial of ''F''[''x''], then it has some root ''a'' and therefore ''p''(''x'') is a multiple of ''x''&nbsp;&minus;&nbsp;''a''. Since ''p''(''x'') is irreducible, this means that ''p''(''x'')&nbsp;=&nbsp;''k''(''x''&nbsp;&minus;&nbsp;''a''), for some ''k''&nbsp;∈&nbsp;''F''&nbsp;\&nbsp;{0}. On the other hand, if ''F'' is not algebraically closed, then there is some non-constant polynomial ''p''(''x'') in ''F''[''x''] without roots in ''F''. Let ''q''(''x'') be some irreducible factor of ''p''(''x''). Since ''p''(''x'') has no roots in ''F'', ''q''(''x'') also has no roots in ''F''. Therefore, ''q''(''x'') has degree greater than one, since every first degree polynomial has one root in ''F''.


===Real numbers===
===Every polynomial is a product of first degree polynomials===
For any [[real number]]&nbsp;{{mvar|x}} the '''absolute value''' or '''modulus''' of&nbsp;{{mvar|x}} is denoted by {{math|{{!}}''x''{{!}}}} (a [[vertical bar]] on each side of the quantity) and is defined as<ref>Mendelson, [http://books.google.com/books?id=A8hAm38zsCMC&pg=PA2 p.&nbsp;2].</ref>
The field ''F'' is algebraically closed if and only if every polynomial ''p''(''x'') of degree ''n''&nbsp;≥&nbsp;1, with [[coefficient]]s in ''F'', [[factorization|splits into linear factors]]. In other words, there are elements ''k'',&nbsp;''x''<sub>1</sub>,&nbsp;''x''<sub>2</sub>,&nbsp;…,&nbsp;''x<sub>n</sub>'' of the field ''F'' such that ''p''(''x'')&nbsp;=&nbsp;''k''(''x''&nbsp;&minus;&nbsp;''x''<sub>1</sub>)(''x''&nbsp;&minus;&nbsp;''x''<sub>2</sub>)&nbsp;···&nbsp;(''x''&nbsp;&minus;&nbsp;''x<sub>n</sub>'').


:<math>|x| = \begin{cases} x, & \mbox{if }  x \ge 0  \\ -x, & \mbox{if } x < 0. \end{cases} </math>
If ''F'' has this property, then clearly every non-constant polynomial in ''F''[''x''] has some root in ''F''; in other words, ''F'' is algebraically closed. On the other hand, that the property stated here holds for ''F'' if ''F'' is algebraically closed follows from the previous property together with the fact that, for any field ''K'', any polynomial in ''K''[''x''] can be written as a product of irreducible polynomials.


As can be seen from the above definition, the absolute value of&nbsp;{{mvar|x}} is always either [[positive number|positive]] or [[0 (number)|zero]], but never [[negative number|negative]].
===Polynomials of prime degree have roots===
J. Shipman showed in 2007 that if every polynomial over ''F'' of prime degree has a root in ''F'', then every non-constant polynomial has a root in ''F'', thus ''F'' is algebraically closed.


From an [[analytic geometry]] point of view, the absolute value of a real number is that number's [[distance]] from zero along the [[real number line]], and more generally the absolute value of the difference of two real numbers is the distance between them. Indeed the notion of an abstract  [[distance function]] in mathematics can be seen to be a generalisation of  the absolute value of the difference (see [[#Distance|"Distance"]] below).
===The field has no proper algebraic extension===
The field ''F'' is algebraically closed if and only if it has no proper [[algebraic extension]].


Since the [[square root]] notation without sign represents the ''positive'' square root, it follows that
If ''F'' has no proper algebraic extension, let ''p''(''x'') be some irreducible polynomial in ''F''[''x'']. Then the [[quotient ring|quotient]] of ''F''[''x''] modulo the [[ideal (ring theory)|ideal]] generated by ''p''(''x'') is an algebraic extension of ''F'' whose [[degree of a field extension|degree]] is equal to the degree of ''p''(''x''). Since it is not a proper extension, its degree is 1 and therefore the degree of ''p''(''x'') is&nbsp;1.


:{|
On the other hand, if ''F'' has some proper algebraic extension ''K'', then the [[Minimal polynomial (field theory)|minimal polynomial]] of an element in ''K''&nbsp;\&nbsp;''F'' is irreducible and its degree is greater than&nbsp;1.
|-
| style="width: 250px" | <math>|a| = \sqrt{a^2}</math>
| ({{EquationRef|1}})
|}


which is sometimes used as a definition of absolute value.<ref>{{Cite book| author=Stewart, James B. | coauthors= | title=Calculus: concepts and contexts | year=2001 | publisher=Brooks/Cole | location=Australia  | isbn=0-534-37718-1 | pages=}}, p.&nbsp;A5</ref>
===The field has no proper finite extension===
The field ''F'' is algebraically closed if and only if it has no finite [[algebraic extension]] because if, within the [[Algebraically closed field#The field has no proper algebraic extension|previous proof]], the word “algebraic” is replaced by the word “finite”, then the proof is still valid.


The absolute value has the following four fundamental properties:
===Every endomorphism of ''F<sup>n</sup>'' has some eigenvector===
The field ''F'' is algebraically closed if and only if, for each natural number ''n'', every [[linear map]] from ''F<sup>n</sup>'' into itself has some [[eigenvector]].


:{|
An endomorphism of ''F<sup>n</sup>'' has an eigenvector if and only if its [[characteristic polynomial]] has some root. Therefore, when ''F'' is algebraically closed, every endomorphism of ''F<sup>n</sup>'' has some eigenvector. On the other hand, if every endomorphism of ''F<sup>n</sup>'' has an eigenvector, let ''p''(''x'') be an element of ''F''[''x'']. Dividing by its leading coefficient, we get another polynomial ''q''(''x'') which has roots if and only if ''p''(''x'') has roots. But if ''q''(''x'')&nbsp;=&nbsp;''x<sup>n</sup>''&nbsp;+&nbsp;''a''<sub>''n''&nbsp;&minus;&nbsp;1</sub>''x''<sup>''n''&nbsp;&minus;&nbsp;1</sup>+&nbsp;···&nbsp;+&nbsp;''a''<sub>0</sub>, then ''q''(''x'') is the characteristic polynomial of the [[companion matrix]]
|-
:<math>\begin{pmatrix}0&0&\cdots&0&-a_0\\1&0&\cdots&0&-a_1\\0&1&\cdots&0&-a_2\\\vdots&\vdots&\ddots&\vdots&\vdots\\0&0&\cdots&1&-a_{n-1}\end{pmatrix}.</math>
| style="width: 250px" |<math>|a| \ge 0 </math>
| style="width: 100px" | ({{EquationRef|2}})
| Non-negativity
|-
|<math>|a| = 0 \iff a = 0 </math>
| ({{EquationRef|3}})
|Positive-definiteness
|-
|<math>|ab| = |a||b|</math>
| ({{EquationRef|4}})
|[[Multiplicativeness]]
|-
|<math>|a+b|  \le |a| + |b|  </math>
| ({{EquationRef|5}})
|[[Subadditivity]]
|}


Other important properties of the absolute value include:
===Decomposition of rational expressions===
The field ''F'' is algebraically closed if and only if every [[rational function]] in one variable ''x'', with coefficients in ''F'', can be written as the sum of a polynomial function with rational functions of the form ''a''/(''x''&nbsp;&minus;&nbsp;''b'')<sup>n</sup>, where ''n'' is a natural number, and ''a'' and ''b'' are elements of ''F''.


:{|
If ''F'' is algebraically closed then, since the irreducible polynomials in ''F''[''x''] are all of degree 1, the property stated above holds by the [[Partial fraction decomposition#Statement of theorem|theorem on partial fraction decomposition]].
|-
| style="width:250px" |<math>|(|a|)| = |a|</math>
| style="width: 100px" | ({{EquationRef|6}})
|[[Idempotence]] (the absolute value of the absolute value is the absolute value)
|-
| style="width:250px" |<math>|-a| = |a|</math>
| style="width: 100px" | ({{EquationRef|7}})
|[[even function|Evenness]] ([[reflection symmetry]] of the graph)
|-
|<math>|a - b| = 0 \iff a = b </math>
| ({{EquationRef|8}})
|[[Identity of indiscernibles]] (equivalent to positive-definiteness)
|-
|<math>|a - b|  \le |a - c| + |c - b|  </math>
| ({{EquationRef|9}})
|[[Triangle inequality]] (equivalent to subadditivity)
|-
|<math>\left|\frac{a}{b}\right| = \frac{|a|}{|b|}\ </math> (if <math>b \ne 0</math>)
| ({{EquationRef|10}})
|Preservation of division (equivalent to multiplicativeness)
|-
|<math>|a-b| \ge |(|a| - |b|)| </math>
| ({{EquationRef|11}})
|(equivalent to subadditivity)
|}


Two other useful properties concerning inequalities are:
On the other hand, suppose that the property stated above holds for the field ''F''. Let ''p''(''x'') be an irreducible element in ''F''[''x'']. Then the rational function 1/''p'' can be written as the sum of a polynomial function ''q'' with rational functions of the form ''a''/(''x''&nbsp;&minus;&nbsp;''b'')<sup>n</sup>. Therefore, the rational expression
:<math>|a| \le b \iff -b \le a \le b </math>
:<math>\frac1{p(x)}-q(x)=\frac{1-p(x)q(x)}{p(x)}</math>
:<math>|a| \ge b \iff a \le -b\ </math> or <math>b \le a </math>
can be written as a quotient of two polynomials in which the denominator is a product of first degree polynomials. Since ''p''(''x'') is irreducible, it must divide this product and, therefore, it must also be a first degree polynomial.


These relations may be used to solve inequalities involving absolute values. For example:
===Relatively prime polynomials and roots===
For any field ''F'', if two polynomials ''p''(''x''),''q''(''x'')&nbsp;∈&nbsp;''F''[''x''] are [[coprime|relatively prime]] then they do not have a common root, for if ''a''&nbsp;∈&nbsp;''F'' was a common root, then&nbsp;''p''(''x'') and &nbsp;''q''(''x'') would both be multiples of ''x''&nbsp;&minus;&nbsp;''a'' and therefore they would not be relatively prime. The fields for which the reverse implication holds (that is, the fields such that whenever two polynomials have no common root then they are relatively prime) are precisely the algebraically closed fields.


:{|
If the field ''F'' is algebraically closed, let ''p''(''x'') and ''q''(''x'') be two polynomials which are not relatively prime and let ''r''(''x'') be their [[greatest common divisor]]. Then, since ''r''(''x'') is not constant, it will have some root ''a'', which will be then a common root of ''p''(''x'') and ''q''(''x'').
|-
|<math>|x-3| \le 9 </math>
|<math>\iff -9 \le x-3 \le 9 </math>
|-
|
|<math>\iff -6 \le x \le 12 </math>
|}


Absolute value is used to define the [[absolute difference]], the standard metric on the real numbers.
If ''F'' is not algebraically closed, let ''p''(''x'') be a polynomial whose degree is at least 1 without roots. Then ''p''(''x'') and ''p''(''x'') are not relatively prime, but they have no common roots (since none of them has roots).


===Complex numbers===
==Other properties==
[[Image:Complex conjugate picture.svg|right|thumb|The absolute value of a complex number&nbsp;{{mvar|z}} is the distance&nbsp;{{mvar|r}} from {{mvar|z}} to the origin. It is also seen in the picture that {{mvar|z}} and its [[complex conjugate]]&nbsp;{{conjugate}} have the same absolute value.]]
If ''F'' is an algebraically closed field and ''n'' is a natural number, then ''F'' contains all ''n''th roots of unity, because these are (by definition) the ''n'' (not necessarily distinct) zeroes of the polynomial ''x<sup>n</sup>''&nbsp;&minus;&nbsp;1. A field extension that is contained in an extension generated by the roots of unity is a ''cyclotomic extension'', and the extension of a field generated by all roots of unity is sometimes called its ''cyclotomic closure''. Thus algebraically closed fields are cyclotomically closed. The converse is not true. Even assuming that every polynomial of the form ''x<sup>n</sup>''&nbsp;&minus;&nbsp;''a'' splits into linear factors is not enough to assure that the field is algebraically closed.


Since the [[complex number]]s are not [[Totally ordered set|ordered]], the definition given above for the real absolute value cannot be directly generalised for a complex number. However the geometric interpretation of the absolute value of a real number as its distance from 0 can be generalised. The absolute value of a complex number is defined as its distance in the [[complex plane]] from the [[origin (mathematics)|origin]] using the [[Pythagorean theorem]]. More generally the absolute value of the difference of two complex numbers is equal to the distance between those two complex numbers.
If a proposition which can be expressed in the language of [[first-order logic]] is true for an algebraically closed field, then it is true for every algebraically closed field with the same [[Characteristic (algebra)|characteristic]]. Furthermore, if such a proposition is valid for an algebraically closed field with characteristic&nbsp;0, then not only is it valid for all other algebraically closed fields with characteristic&nbsp;0, but there is some natural number ''N'' such that the proposition is valid for every algebraically closed field with characteristic&nbsp;''p'' when ''p''&nbsp;&gt;&nbsp;''N''.<ref>See subsections ''Rings and fields'' and ''Properties of mathematical theories'' in §2 of J. Barwise's "An introduction to first-order logic".</ref>


For any complex number
Every field ''F'' has some extension which is algebraically closed. Among all such extensions there is one and ([[Up to|up to isomorphism]], but not [[essentially unique|unique isomorphism]]) only one which is an [[algebraic extension]] of ''F'';<ref>See Lang's ''Algebra'', §VII.2 or van der Waerden's ''Algebra I'', §10.1.</ref> it is called the [[algebraic closure]] of ''F''.


:<math>z = x + iy,</math>
The theory of algebraically closed fields has [[quantifier elimination]].
 
where {{mvar|x}} and {{mvar|y}} are real numbers, the '''absolute value''' or '''modulus''' of&nbsp;{{mvar|z}} is denoted {{math|{{!}}''z''{{!}}}} and is given by<ref>{{cite book|author=González, Mario O.|title=Classical Complex Analysis|publisher=CRC Press|year=1992|isbn=9780824784157|page=19|url=http://books.google.com/books?id=ncxL7EFr7GsC&pg=PA19}}</ref>
 
:<math>|z| =  \sqrt{x^2 + y^2}.</math>
 
When  the complex part {{mvar|y}} is zero this is the same as the absolute value of the real number&nbsp;{{mvar|x}}.
 
When a complex number&nbsp;{{mvar|z}} is expressed in [[Complex number#Polar form|polar form]] as
 
:<math>z = r e^{i \theta}</math>
 
with {{math|''r'' ≥ 0}} and θ real, its absolute value is
 
:<math>|z| = r</math>.
 
The absolute value of a complex number can be written in the complex analogue of equation&nbsp;[[#math 1|(1)]] above as:
 
:<math>|z| = \sqrt{z \cdot \overline{z}}</math>
 
where <math>\overline z</math> is the [[complex conjugate]] of&nbsp;{{mvar|z}}.
The complex absolute value shares all the properties of the real absolute value given in equations&nbsp;[[#math 2|(2)–(11)]] above.
 
Since the positive reals form a subgroup of the complex numbers under multiplication, we may think of absolute value as an [[endomorphism]] of the [[multiplicative group]] of the complex numbers.<ref>{{citation
| last = Lorenz | first = Falko
| isbn = 978-0-387-72487-4
| location = New York
| mr = 2371763
| page = 39
| publisher = Springer
| series = Universitext
| title = Algebra. Vol. II. Fields with structure, algebras and advanced topics
| doi = 10.1007/978-0-387-72488-1
| year = 2008}}.</ref>
 
==Absolute value function==
[[Image:Absolute value.svg|thumb|360px|The [[graph of a function|graph]] of the absolute value function for real numbers]]
[[Image:Absolute value composition.svg|256px|thumb|[[composition of functions|Composition]] of absolute value with a [[cubic function]] in different orders]]
The real absolute value function is [[continuous function|continuous]] everywhere. It is [[derivative|differentiable]] everywhere except for {{mvar|x}}&nbsp;=&nbsp;0.  It is [[monotonic function|monotonically decreasing]] on the interval {{open-closed|−∞,0}} and monotonically increasing on the interval {{closed-open|0,+∞}}. Since a real number and its [[additive inverse|opposite]] have the same absolute value, it is an [[even function]], and is hence not [[invertible]].
 
Both the real and complex functions are [[idempotent]].
 
It is a [[piecewise linear function|piecewise linear]], [[convex function|convex]] function.
 
===Relationship to the sign function===
The absolute value function of a real number returns its value irrespective of its sign, whereas the [[sign function|sign (or signum) function]] returns a number's sign irrespective of its value. The following equations show the relationship between these two functions:
:<math>|x| = x \sgn(x),</math>
or
:<math> |x| \sgn(x) = x,</math>
and for {{math|''x'' ≠ 0}},
:<math>\sgn(x) = \frac{|x|}{x}.</math>
 
===Derivative===
The real absolute value function has a derivative for every {{math|''x'' ≠ 0}}, but is not [[differentiable function|differentiable]] at {{math|1=''x'' = 0}}. Its derivative for {{math|''x'' ≠ 0}} is given by the [[step function]]<ref name="MathWorld">[http://mathworld.wolfram.com/AbsoluteValue.html Weisstein, Eric W. ''Absolute Value.'' From MathWorld – A Wolfram Web Resource.]</ref><ref name="BS163">Bartel and Sherbert, p.&nbsp;163</ref>
:<math>\frac{d|x|}{dx} = \frac{x}{|x|} = \begin{cases} -1 & x<0 \\  1 & x>0. \end{cases}</math>
 
The [[subderivative|subdifferential]] of&nbsp;{{math|{{!}}''x''{{!}}}} at&nbsp;{{math|1=''x'' = 0}} is the [[interval (mathematics)|interval]]&nbsp;{{closed-closed|−1,1}}.<ref>Peter Wriggers, Panagiotis Panatiotopoulos, eds., ''New Developments in Contact Problems'', 1999, ISBN 3-211-83154-1, [http://books.google.com/books?id=tiBtC4GmuKcC&pg=PA31 p.&nbsp;31–32]</ref>
 
The [[complex number|complex]] absolute value function is continuous everywhere but [[complex differentiable]] ''nowhere'' because it violates the [[Cauchy–Riemann equations]].<ref name="MathWorld"/>
 
The second derivative of&nbsp;{{math|{{!}}''x''{{!}}}} with respect to&nbsp;{{mvar|x}} is zero everywhere except zero, where it does not exist. As a [[generalised function]], the second derivative may be taken as two times the [[Dirac delta function]].
 
===Antiderivative===
The [[antiderivative]] (indefinite integral) of the absolute value function is
 
:<math>\int|x|dx=\frac{x|x|}{2}+C,</math>
 
where {{mvar|C}} is an arbitrary [[constant of integration]].
 
==Distance==
{{see also|Metric space}}
The absolute value is closely related to the idea of distance. As noted above, the absolute value of a real or complex number is the [[distance]] from that number to the origin, along the real number line, for real numbers, or in the complex plane, for complex numbers, and more generally, the absolute value of the difference of two real or complex numbers is the distance between them.
 
The standard [[Euclidean distance]] between two points
 
:<math>a = (a_1, a_2, \dots , a_n) </math>
 
and
 
:<math>b = (b_1, b_2, \dots , b_n) </math>
 
in [[Euclidean space|Euclidean {{mvar|n}}-space]] is defined as:
:<math>\sqrt{\sum_{i=1}^n(a_i-b_i)^2}. </math>
 
This can be seen to be a generalisation of {{math|{{!}}''a'' − ''b''{{!}}}}, since if {{mvar|a}} and {{mvar|b}} are real, then by [[#math 1|equation&nbsp;(1)]],
:<math>|a - b| = \sqrt{(a - b)^2}.</math>
 
While if
 
:<math> a = a_1 + i a_2 </math>
 
and
 
:<math> b = b_1 + i b_2 </math>
 
are complex numbers, then
 
:{|
|-
|<math>|a - b| </math>
|<math> = |(a_1 + i a_2) - (b_1 + i b_2)|</math>
|-
|
|<math> = |(a_1 - b_1) + i(a_2 - b_2)|</math>
|-
|
|<math> = \sqrt{(a_1 - b_1)^2 + (a_2 - b_2)^2}.</math>
|}
 
The above shows that the "absolute value" distance for the real numbers or the complex numbers, agrees with the standard Euclidean distance they inherit as a result of considering them as the one and two-dimensional Euclidean spaces respectively.
 
The properties of the absolute value of the difference of two real or complex numbers: non-negativity, identity of indiscernibles, symmetry and the triangle inequality given above, can be seen to motivate the more general notion of a [[distance function]] as follows:
 
A real valued function {{mvar|d}} on a set {{math|''X'' × ''X''}} is called a [[Metric (mathematics)|metric]] (or a ''distance function'') on&nbsp;{{mvar|X}}, if it satisfies the following four axioms:<ref>These axioms are not minimal; for instance, non-negativity can be derived from the other three: {{math|1=0 = ''d''(''a'', ''a'') ≤ ''d''(''a'', ''b'') + ''d''(''b'', ''a'') = 2''d''(''a'', ''b'')}}.</ref>
:{|
|-
|style="width:250px" | <math>d(a, b) \ge 0 </math>
|Non-negativity
|-
|<math>d(a, b) = 0 \iff a = b </math>
|Identity of indiscernibles
|-
|<math>d(a, b) = d(b, a) </math>
|Symmetry
|-
|<math>d(a, b)  \le d(a, c) + d(c, b)  </math>
|Triangle inequality
|}
 
==Generalizations==
 
===Ordered rings===
The definition of absolute value given for real numbers above can be extended to any [[ordered ring]]. That is, if&nbsp;{{mvar|a}} is an element of an ordered ring&nbsp;''R'', then the '''absolute value''' of&nbsp;{{mvar|a}}, denoted by {{math|{{!}}''a''{{!}}}}, is defined to be:<ref>Mac Lane, [http://books.google.com/books?id=L6FENd8GHIUC&pg=PA264 p.&nbsp;264].</ref>
 
:<math>|a| = \begin{cases} a, & \mbox{if }  a \ge 0  \\ -a,  & \mbox{if } a \le 0 \end{cases} \; </math>
 
where {{math|−''a''}} is the [[additive inverse]] of&nbsp;{{mvar|a}}, and 0 is the additive [[identity element]].
 
===Fields===
{{main|Absolute value (algebra)}}
The fundamental properties of the absolute value for real numbers given in [[#math 2|(2)]]–(5) above, can be used to generalise the notion of absolute value to an arbitrary field, as follows.
 
A real-valued function&nbsp;{{mvar|v}} on a [[field (mathematics)|field]]&nbsp;{{mvar|F}} is called an ''absolute value'' (also a ''modulus'', ''magnitude'', ''value'', or ''valuation'')<ref>Shechter, [http://books.google.com/books?id=eqUv3Bcd56EC&pg=PA260 p.&nbsp;260]. This meaning of ''valuation'' is rare. Usually, a [[valuation (algebra)|valuation]] is the logarithm of the inverse of an absolute value</ref> if it satisfies the following four axioms:
 
:{| cellpadding=10
|-
|<math>v(a) \ge 0 </math>
|Non-negativity
|-
|<math>v(a) = 0 \iff a = \mathbf{0} </math>
|Positive-definiteness
|-
|<math>v(ab) = v(a) v(b) </math>
|Multiplicativeness
|-
|<math>v(a+b)  \le v(a) + v(b)  </math>
|Subadditivity or the triangle inequality
|}
 
Where '''0''' denotes the [[additive identity]] element of&nbsp;{{mvar|F}}. It follows from positive-definiteness and multiplicativeness that {{math|1=''v''('''1''') = 1}}, where '''1''' denotes the [[multiplicative identity]] element of&nbsp;{{mvar|F}}. The real and complex absolute values defined above are examples of absolute values for an arbitrary field.
 
If {{mvar|v}} is an absolute value on&nbsp;{{mvar|F}}, then the function&nbsp;{{mvar|d}} on {{math|''F'' × ''F''}}, defined by {{math|1=''d''(''a'', ''b'') = ''v''(''a'' − ''b'')}}, is a metric and the following are equivalent:
 
* {{mvar|d}} satisfies the [[ultrametric]] inequality <math>d(x, y) \leq \max(d(x,z),d(y,z))</math> for all {{mvar|x}}, {{mvar|y}}, {{mvar|z}} in&nbsp;{{mvar|F}}.
 
* <math> \big\{ v\Big({\textstyle \sum_{k=1}^n } \mathbf{1}\Big) : n \in \mathbb{N} \big\} </math> is [[bounded set|bounded]] in&nbsp;'''R'''.
 
* <math> v\Big({\textstyle \sum_{k=1}^n } \mathbf{1}\Big) \le 1\ </math> for every <math>n \in \mathbb{N}.</math>
 
* <math> v(a) \le 1 \Rightarrow v(1+a) \le 1\ </math> for all <math>a \in F.</math>
 
* <math> v(a + b) \le \mathrm{max}\{v(a), v(b)\}\ </math> for all <math>a, b \in F.</math>
 
An absolute value which satisfies any (hence all) of the above conditions is said to be '''non-Archimedean''', otherwise it is said to be [[Archimedean field|Archimedean]].<ref>Shechter, [http://books.google.com/books?id=eqUv3Bcd56EC&pg=PA260 pp.&nbsp;260–261].</ref>
 
===Vector spaces===
{{Main|Norm (mathematics)}}
Again the fundamental properties of the absolute value for real numbers can be used, with a slight modification, to generalise the notion to an arbitrary vector space.
 
A real-valued function on a [[vector space]]&nbsp;{{mvar|V}} over a field&nbsp;{{mvar|F}}, represented as <big>{{math|‖·<!-- do not replace with &sdot;: not a multiplication sign! -->‖}}</big>, is called an '''absolute value''', but more usually a [[Norm (mathematics)|'''norm''']], if it satisfies the following axioms:
 
For all&nbsp;{{mvar|a}} in&nbsp;{{mvar|F}}, and {{math|'''v'''}}, {{math|'''u'''}} in&nbsp;{{mvar|V}},
 
:{| cellpadding=10
|-
|<math>\|\mathbf{v}\|  \ge 0 </math>
|Non-negativity
|-
|<math>\|\mathbf{v}\| = 0 \iff \mathbf{v} = 0</math>
|Positive-definiteness
|-
|<math>\|a \mathbf{v}\| = |a| \|\mathbf{v}\| </math>
|Positive homogeneity or positive scalability
|-
|<math>\|\mathbf{v} + \mathbf{u}\| \le \|\mathbf{v}\| + \|\mathbf{u}\| </math>
|Subadditivity or the triangle inequality
|}
 
The norm of a vector is also called its ''length'' or ''magnitude''.
 
In the case of [[Euclidean space]]&nbsp;{{math|'''R'''<sup>''n''</sup>}}, the function defined by
 
:<math>\|(x_1, x_2, \dots , x_n) \| = \sqrt{\sum_{i=1}^{n} x_i^2}</math>
 
is a norm called the [[Euclidean norm]]. When the real numbers&nbsp;{{math|'''R'''}} are considered as the one-dimensional vector space&nbsp;{{math|'''R'''<sup>1</sup>}}, the absolute value is a [[Norm (mathematics)|norm]], and is the {{mvar|p}}-norm (see [[L^p_space#Definition|L<sup>p</sup> space]]) for any&nbsp;{{mvar|p}}. In fact the absolute value is the "only" norm on {{math|'''R'''<sup>1</sup>}}, in the sense that, for every norm {{math|‖·‖}} on&nbsp;{{math|'''R'''<sup>1</sup>}}, {{math|1=‖''x''‖ = ‖1‖ ⋅ {{!}}''x''{{!}}}}. The complex absolute value is a special case of the norm in an [[inner product space]]. It is identical to the Euclidean norm, if the [[complex plane]] is identified with the [[Euclidean plane]]&nbsp;{{math|'''R'''<sup>2</sup>}}.


==Notes==
==Notes==
Line 317: Line 67:


==References==
==References==
* Bartle; Sherbert; ''Introduction to real analysis'' (4th ed.), John Wiley & Sons, 2011 ISBN 978-0-471-43331-6.
* {{Citation | last = Barwise | first = Jon | author-link = Jon Barwise | year = 1978 | contribution = An introduction to first-order logic | editor-last = Barwise | editor-first = Jon | title = Handbook of Mathematical Logic | series = Studies in Logic and the Foundations of Mathematics | publisher = North Holland | isbn = 0-7204-2285-X}}
* Nahin, Paul J.; [http://www.amazon.com/gp/reader/0691027951 ''An Imaginary Tale'']; Princeton University Press; (hardcover, 1998). ISBN 0-691-02795-1.
* {{Lang Algebra}}
* Mac Lane, Saunders, Garrett Birkhoff, ''Algebra'', American Mathematical Soc., 1999. ISBN 978-0-8218-1646-2.
* {{citation|last = Shipman|first = Joseph|year = 2007|title = Improving the Fundamental Theorem of Algebra|periodical = Mathematical Intelligencer|volume = 29|issue = 4|pages = 9–14|doi=10.1007/BF02986170|issn = 0343-6993}}
* Mendelson, Elliott, ''Schaum's Outline of Beginning Calculus'', McGraw-Hill Professional, 2008. ISBN 978-0-07-148754-2.
* {{Citation | last = van der Waerden | first = Bartel Leendert | author-link = Bartel Leendert van der Waerden | title = Algebra | volume = I | edition = 7th | year = 2003 | publisher = Springer-Verlag | isbn = 0-387-40624-7}}
* O'Connor, J.J. and Robertson, E.F.; [http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Argand.html "Jean Robert Argand"].
* Schechter, Eric; ''Handbook of Analysis and Its Foundations'', pp.&nbsp;259–263, [http://books.google.com/books?id=eqUv3Bcd56EC&pg=PA259 "Absolute Values"],  Academic Press (1997) ISBN 0-12-622760-8.
 
==External links==
* {{springer|title=Absolute value|id=p/a010370}}
* {{PlanetMath | urlname=AbsoluteValue | title=absolute value | id=448}}
* {{MathWorld | urlname=AbsoluteValue | title=Absolute Value}}


{{DEFAULTSORT:Absolute Value}}
{{DEFAULTSORT:Algebraically Closed Field}}
[[Category:Special functions]]
[[Category:Abstract algebra]]
[[Category:Field theory]]

Revision as of 03:09, 28 June 2013

30 year-old Entertainer or Range Artist Wesley from Drumheller, really loves vehicle, property developers properties for sale in singapore singapore and horse racing. Finds inspiration by traveling to Works of Antoni Gaudí. In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F[x], the ring of polynomials in the variable x with coefficients in F.

Examples

As an example, the field of real numbers is not algebraically closed, because the polynomial equation x2 + 1 = 0  has no solution in real numbers, even though all its coefficients (1 and 0) are real. The same argument proves that no subfield of the real field is algebraically closed; in particular, the field of rational numbers is not algebraically closed. Also, no finite field F is algebraically closed, because if a1, a2, …, an are the elements of F, then the polynomial (x − a1)(x − a2) ··· (x − an) + 1 has no zero in F. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed. Another example of an algebraically closed field is the field of (complex) algebraic numbers.

Equivalent properties

Given a field F, the assertion “F is algebraically closed” is equivalent to other assertions:

The only irreducible polynomials are those of degree one

The field F is algebraically closed if and only if the only irreducible polynomials in the polynomial ring F[x] are those of degree one.

The assertion “the polynomials of degree one are irreducible” is trivially true for any field. If F is algebraically closed and p(x) is an irreducible polynomial of F[x], then it has some root a and therefore p(x) is a multiple of x − a. Since p(x) is irreducible, this means that p(x) = k(x − a), for some k ∈ F \ {0}. On the other hand, if F is not algebraically closed, then there is some non-constant polynomial p(x) in F[x] without roots in F. Let q(x) be some irreducible factor of p(x). Since p(x) has no roots in F, q(x) also has no roots in F. Therefore, q(x) has degree greater than one, since every first degree polynomial has one root in F.

Every polynomial is a product of first degree polynomials

The field F is algebraically closed if and only if every polynomial p(x) of degree n ≥ 1, with coefficients in F, splits into linear factors. In other words, there are elements k, x1, x2, …, xn of the field F such that p(x) = k(x − x1)(x − x2) ··· (x − xn).

If F has this property, then clearly every non-constant polynomial in F[x] has some root in F; in other words, F is algebraically closed. On the other hand, that the property stated here holds for F if F is algebraically closed follows from the previous property together with the fact that, for any field K, any polynomial in K[x] can be written as a product of irreducible polynomials.

Polynomials of prime degree have roots

J. Shipman showed in 2007 that if every polynomial over F of prime degree has a root in F, then every non-constant polynomial has a root in F, thus F is algebraically closed.

The field has no proper algebraic extension

The field F is algebraically closed if and only if it has no proper algebraic extension.

If F has no proper algebraic extension, let p(x) be some irreducible polynomial in F[x]. Then the quotient of F[x] modulo the ideal generated by p(x) is an algebraic extension of F whose degree is equal to the degree of p(x). Since it is not a proper extension, its degree is 1 and therefore the degree of p(x) is 1.

On the other hand, if F has some proper algebraic extension K, then the minimal polynomial of an element in K \ F is irreducible and its degree is greater than 1.

The field has no proper finite extension

The field F is algebraically closed if and only if it has no finite algebraic extension because if, within the previous proof, the word “algebraic” is replaced by the word “finite”, then the proof is still valid.

Every endomorphism of Fn has some eigenvector

The field F is algebraically closed if and only if, for each natural number n, every linear map from Fn into itself has some eigenvector.

An endomorphism of Fn has an eigenvector if and only if its characteristic polynomial has some root. Therefore, when F is algebraically closed, every endomorphism of Fn has some eigenvector. On the other hand, if every endomorphism of Fn has an eigenvector, let p(x) be an element of F[x]. Dividing by its leading coefficient, we get another polynomial q(x) which has roots if and only if p(x) has roots. But if q(x) = xn + an − 1xn − 1+ ··· + a0, then q(x) is the characteristic polynomial of the companion matrix

Decomposition of rational expressions

The field F is algebraically closed if and only if every rational function in one variable x, with coefficients in F, can be written as the sum of a polynomial function with rational functions of the form a/(x − b)n, where n is a natural number, and a and b are elements of F.

If F is algebraically closed then, since the irreducible polynomials in F[x] are all of degree 1, the property stated above holds by the theorem on partial fraction decomposition.

On the other hand, suppose that the property stated above holds for the field F. Let p(x) be an irreducible element in F[x]. Then the rational function 1/p can be written as the sum of a polynomial function q with rational functions of the form a/(x − b)n. Therefore, the rational expression

can be written as a quotient of two polynomials in which the denominator is a product of first degree polynomials. Since p(x) is irreducible, it must divide this product and, therefore, it must also be a first degree polynomial.

Relatively prime polynomials and roots

For any field F, if two polynomials p(x),q(x) ∈ F[x] are relatively prime then they do not have a common root, for if a ∈ F was a common root, then p(x) and  q(x) would both be multiples of x − a and therefore they would not be relatively prime. The fields for which the reverse implication holds (that is, the fields such that whenever two polynomials have no common root then they are relatively prime) are precisely the algebraically closed fields.

If the field F is algebraically closed, let p(x) and q(x) be two polynomials which are not relatively prime and let r(x) be their greatest common divisor. Then, since r(x) is not constant, it will have some root a, which will be then a common root of p(x) and q(x).

If F is not algebraically closed, let p(x) be a polynomial whose degree is at least 1 without roots. Then p(x) and p(x) are not relatively prime, but they have no common roots (since none of them has roots).

Other properties

If F is an algebraically closed field and n is a natural number, then F contains all nth roots of unity, because these are (by definition) the n (not necessarily distinct) zeroes of the polynomial xn − 1. A field extension that is contained in an extension generated by the roots of unity is a cyclotomic extension, and the extension of a field generated by all roots of unity is sometimes called its cyclotomic closure. Thus algebraically closed fields are cyclotomically closed. The converse is not true. Even assuming that every polynomial of the form xn − a splits into linear factors is not enough to assure that the field is algebraically closed.

If a proposition which can be expressed in the language of first-order logic is true for an algebraically closed field, then it is true for every algebraically closed field with the same characteristic. Furthermore, if such a proposition is valid for an algebraically closed field with characteristic 0, then not only is it valid for all other algebraically closed fields with characteristic 0, but there is some natural number N such that the proposition is valid for every algebraically closed field with characteristic p when p > N.[1]

Every field F has some extension which is algebraically closed. Among all such extensions there is one and (up to isomorphism, but not unique isomorphism) only one which is an algebraic extension of F;[2] it is called the algebraic closure of F.

The theory of algebraically closed fields has quantifier elimination.

Notes

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References

  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Template:Lang Algebra
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  1. ↑ See subsections Rings and fields and Properties of mathematical theories in §2 of J. Barwise's "An introduction to first-order logic".
  2. ↑ See Lang's Algebra, §VII.2 or van der Waerden's Algebra I, §10.1.
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