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| {{about|the ring of complex numbers integral over {{math|ℤ}}|the general notion of algebraic integer|Integrality}}
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| {{Distinguish|algebraic element|algebraic number}}
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| In [[number theory]], an '''algebraic integer''' is a [[complex number]] that is a [[root of a function|root]] of some [[monic polynomial]] (a polynomial whose leading coefficient is 1) with coefficients in {{math|ℤ}} (the set of [[integer]]s). The set of all algebraic integers is closed under addition and multiplication and therefore is a [[subring]] of complex numbers denoted by '''A'''. The ring '''A''' is the [[integral closure]] of regular integers {{math|ℤ}} in complex numbers.
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| The [[ring of integers]] of a [[number field]] ''K'', denoted by ''O<sub>K</sub>'', is the intersection of ''K'' and '''A''': it can also be characterised as the maximal [[Order (ring theory)|order]] of the field ''K''.
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| Each algebraic integer belongs to the ring of integers of some number field. A number ''x'' is an algebraic integer [[if and only if]] the ring {{math|ℤ}}[''x''] is [[Finitely generated group|finitely generated]] as an [[abelian group]], which is to say, as a [[Free Z-module|{{math|ℤ}}-module]].
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| ==Definitions==
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| The following are equivalent definitions of an algebraic integer. Let ''K'' be a [[number field]] (i.e., a [[finite extension]] of <math>\mathbb Q</math>, the set of [[rational number]]s), in other words, <math>K = \mathbb{Q}(\theta)</math> for some algebraic number <math>\theta \in \mathbb{C}</math> by the [[primitive element theorem]].
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| * <math>\alpha \in K</math> is an algebraic integer if there exists a monic polynomial <math>f(x) \in \mathbb{Z}[x]</math> such that <math>f(\alpha) = 0</math>.
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| * <math>\alpha \in K</math> is an algebraic integer if the minimal monic polynomial of <math>\alpha</math> over <math>\mathbb Q</math> is in <math>\mathbb{Z}[x]</math>.
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| * <math>\alpha \in K</math> is an algebraic integer if <math>\mathbb{Z}[\alpha]</math> is a finitely generated <math>\mathbb Z</math>-module.
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| * <math>\alpha \in K </math> is an algebraic integer if there exists a finitely generated <math>\mathbb{Z}</math>-submodule <math>M \subset \mathbb{C}</math> such that <math>\alpha M \subseteq M</math>.
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| Algebraic integers are a special case of [[integral element]]s of a ring extension. In particular, an algebraic integer is an integral element of a finite extension <math>K / \mathbb{Q}</math>.
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| ==Examples==
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| * The only algebraic integers which are found in the set of [[rational numbers]] are the integers. In other words, the intersection of '''Q''' and '''A''' is exactly '''Z'''. The rational number ''a''/''b'' is not an algebraic integer unless ''b'' divides ''a''. Note that the leading coefficient of the polynomial ''bx'' − ''a'' is the integer ''b''. As another special case, the square root √''n'' of a non-negative integer ''n'' is an algebraic integer, and so is irrational unless ''n'' is a [[square number|perfect square]].
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| *If ''d'' is a [[square free]] integer then the extension ''K'' = '''Q'''(√{{overline|''d''}}) is a [[quadratic field extension|quadratic field]] of rational numbers. The ring of algebraic integers ''O<sub>K</sub>'' contains √{{overline|''d''}} since this is a root of the monic polynomial ''x''<sup>2</sup> − ''d''. Moreover, if ''d'' ≡ 1 (mod 4) the element (1 + √{{overline|''d''}})/2 is also an algebraic integer. It satisfies the polynomial ''x''<sup>2</sup> − ''x'' + (1 − ''d'')/4 where the [[constant term]] (1 − ''d'')/4 is an integer. The full ring of integers is generated by √{{overline|''d''}} or (1 + √{{overline|''d''}})/2 respectively. See [[quadratic integer]]s for more.
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| *The ring of integers of the field <math>F = \mathbf Q[\alpha], \alpha = \sqrt[3] m </math> has the following [[integral basis]], writing <math>m = hk^2</math> for two square-free coprime integers ''h'' and ''k'':<ref>{{Citation | last1=Marcus | first1=Daniel A. | title=Number fields | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90279-1 | year=1977}}, chapter 2, p. 38 and exercise 41.</ref>
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| :<math>\begin{cases}
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| 1, \alpha, \frac{\alpha^2 \pm k^2 \alpha + k^2}{3k} & m \equiv \pm 1 \mod 9 \\
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| 1, \alpha, \frac{\alpha^2}k & \mathrm{else}
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| \end{cases}</math>
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| * If ''ζ<sub>n</sub>'' is a primitive ''n''-th [[root of unity]], then the ring of integers of the [[cyclotomic field]] '''Q'''(''ζ<sub>n</sub>'') is precisely '''Z'''[''ζ<sub>n</sub>''].
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| * If ''α'' is an algebraic integer then <math>\beta=\sqrt[n]{\alpha}</math> is another algebraic integer. A polynomial for ''β'' is obtained by substituting ''x''<sup>''n''</sup> in the polynomial for ''α''.
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| ==Non-example==
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| * If ''P''(''x'') is a [[Primitive polynomial (ring theory)|primitive polynomial]] which has integer coefficients but is not monic, and ''P'' is [[irreducible polynomial|irreducible]] over '''Q''', then none of the roots of ''P'' are algebraic integers. (Here ''primitive'' is used in the sense that the [[highest common factor]] of the set of coefficients of ''P'' is 1; this is weaker than requiring the coefficients to be pairwise relatively prime.)
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| ==Facts==
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| * The sum, difference and product of two algebraic integers is an algebraic integer. In general their quotient is not. The monic polynomial involved is generally of higher [[degree of a polynomial|degree]] than those of the original algebraic integers, and can be found by taking resultants and factoring. For example, if ''x''<sup>2</sup> − ''x'' − 1 = 0, ''y''<sup>3</sup> − ''y'' − 1 = 0 and ''z'' = ''xy'', then eliminating ''x'' and ''y'' from ''z'' − ''xy'' and the polynomials satisfied by ''x'' and ''y'' using the [[resultant]] gives ''z''<sup>6</sup> − 3''z''<sup>4</sup> − 4''z''<sup>3</sup> + ''z''<sup>2</sup> + ''z'' − 1, which is irreducible, and is the monic polynomial satisfied by the product. (To see that the ''xy'' is a root of the x-resultant of ''z'' − ''xy'' and ''x''<sup>2</sup> − ''x'' − 1, one might use the fact that the resultant is contained in the ideal generated by its two input polynomials.)
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| * Any number constructible out of the integers with roots, addition, and multiplication is therefore an algebraic integer; but not all algebraic integers are so constructible: in a naïve sense, most roots of irreducible [[quintic]]s are not. This is the [[Abel-Ruffini theorem]]. <!-- what is the meaning of "most" roots of irreducible quintics? By counting, there are as many non-solvable as solvable quintics. Are coefficients of the quintic taken "randomly" from the integers? There ain't no such "random" integer! //--><!--How about this: Consider irreducible quintics of degree n, with integer coefficients with absolute value <= a. Does the proportion of them that are solvable not approach 0 as n and a go to infinitely, whether separately or together?-->
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| * Every root of a monic polynomial whose coefficients are algebraic integers is itself an algebraic integer. In other words, the algebraic integers form a ring which is [[integrally closed]] in any of its extensions.
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| * The ring of algebraic integers '''A''' is a [[Bézout domain]].
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| ==References==
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| <references />
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| * Daniel A. Marcus, ''Number Fields'', third edition, Springer-Verlag, 1977
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| ==See also==
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| *[[Gaussian integer]]
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| *[[Eisenstein integer]]
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| *[[Root of unity]]
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| *[[Dirichlet's unit theorem]]
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| *[[fundamental unit (number theory)|Fundamental units]]
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| [[Category:Algebraic numbers]]
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| [[Category:Integers]]
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Gardeners in order to spend as much time as they are able preening their garden. Is actually one in order to keep your backyard safe and clean. I'm going to show you how to create special gaden walling. Will certainly keep the animals like rabbits and deer by means of precious garden area. It also keeps people from walking all over-the-counter garden.
When you consider landscaping, consider your water needs depending upon the local local climate. These factors will help you to determine which to plant. For example, if your area has water limitations or restrictions, then you want to choose low maintenance plants that require little watering.
Another thing is your shrubbery. An individual might be going to want to trim your trees, hedges and vines around your house to selected everything is clean-cut. Therefore be qualified for have the debris all wheeled away in a wheel barrow and taken away along with landscapers. All you need will done nicely.
The mixture of talent and experience that's not a problem youth and potential for this LSU team seems regarding coming every other. It's hard to say that really clean nothing else that could be explained along with first five games. The talent on this team got them past Washington and ULL. The experience of knowing the best way to win got them by Vanderbilt and Mississippi City. It took a little more in Athens and it may take more this weekend against The carolinas. The Tigers couldn't be in a better spot than where they sit at the moment. It hasn't been pretty irritated hasn't been easy but LSU is 5-0. Earlier guys know what it takes and younger guys are starting to take shape. It's going to be fun; heart-stopping fun this weekend in Death Valley.
It is also vital that you simply pull out and treat all weeds as soon as capable. For this reason horse owners need a weed control software. Your paddock should be sprayed with safe modern pesticides that tackle the weeds discovered by the soil analysis.
Here's my analysis of the I learnt from hearing Bill Clinton in person and noting how he was offered. You should be able to evolve at least some among these points to fit your own examples.
A proper inspection for this trees accomplished and then your decision to fell a tree is taken. Every person preferred to retain the tree pleasant. In congested areas, it requires be done very carefully to avoid any damage of the buildings, structures or anything much more situated near. It is done by felling the tree in sections, using ropes, pulleys and winches.
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