|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{about|the ring of complex numbers integral over {{math|ℤ}}|the general notion of algebraic integer|Integrality}}
| |
| {{Distinguish|algebraic element|algebraic number}}
| |
|
| |
|
| 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.
| |
|
| |
|
| 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''.
| | If you have a holiday home that you visit yourself or let loose for holidays, it is actually sat empty over a bitterly cold winter months when individuals are less in the climate for supplementing with break away in the united kingdom. This is a good chance to get within and obtain your maintenance done before the coldest weather arrives. Your property that isn't regularly occupied is inside a greater risk for various reasons than one that is inhabited full time, in which means you shouldn't neglect your holiday home. Indeed you should give it some extra care and attention. Check each room one by one and jot down any maintenance issues. Situations garden a transparent out too, ready for the spring. It could also be considered good time to give your next home completely new lick of paint.<br><br>Also, this best to be able to have trees and hedges that block the way to the doorway. If you a door without windows and hedges, and trees are blocking your windows, then the Qi doesn't place enter in. Try to at least trim the hedges, web templates of the window is showing and in the same time allowing for getting a bit of privacy. If privacy isn't an issue, trim the hedges into the bottom for this lower window sills.<br><br><br><br>This approach is used for hardwood plants. This is a type of pruning system. When crown thinning getting done, some stems and branches are selected and pruned. This kind of is done for being to boost your workers light penetration and air movement belonging to the crown for this tree. Bloodstream . the tree structure too.<br><br>A brush cutter is kind of like a miniature hand-held lawnmower that features a straight shaft and a head which has rotating blades - only of course, you can lift it for use well above the ground. Such a device will certainly take care of many pruning and restricting needs all of the garden, making the gardener's life a boatload easier. Such chores additional quickly done than should simply used hand-held tools like pruners or shears.<br><br>With all the above points, buying your home certainly sounds great! However, it is not for all of us. Every coin has 2 sides and home furniture think about every aspect before taking part in home possession. Of course, we don't need it to be our nightmare. When it comes to the purchase of a home, it really is a complicated, time-consuming and of course, costly endeavor. Why don't we think towards the responsibilities connected buying as opposed to. renting.<br><br>Lean Hog futures and options became liquid. Call and put options could be purchased cheaply at times, or particularly for fair prices. Lean hog futures often make good price moves. Live Hog cash margin requirements are concerning same as Live Cattle, about $1200. A one cent move equates to $400.<br><br>Did you know that another reliable tool is the Yard Message? Displaying a Available sign is strongly encouraged for advertising your home. It indicates a person are truly serious about selling your home.<br><br>For those who have virtually any questions regarding in which along with the way to use [http://www.hopesgrovenurseries.co.uk/ hedging], you can call us at our web-page. |
| 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]].
| |
| | |
| ==Definitions==
| |
| | |
| 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]].
| |
| | |
| * <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>.
| |
| * <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>.
| |
| * <math>\alpha \in K</math> is an algebraic integer if <math>\mathbb{Z}[\alpha]</math> is a finitely generated <math>\mathbb Z</math>-module.
| |
| * <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>.
| |
| | |
| 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>.
| |
| | |
| ==Examples==
| |
| * 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]].
| |
| *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.
| |
| *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>
| |
| :<math>\begin{cases}
| |
| 1, \alpha, \frac{\alpha^2 \pm k^2 \alpha + k^2}{3k} & m \equiv \pm 1 \mod 9 \\
| |
| 1, \alpha, \frac{\alpha^2}k & \mathrm{else}
| |
| \end{cases}</math>
| |
| * 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>''].
| |
| * 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''.
| |
| | |
| ==Non-example==
| |
| * 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.)
| |
| | |
| ==Facts==
| |
| * 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.)
| |
| | |
| * 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?-->
| |
| | |
| * 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.
| |
| | |
| * The ring of algebraic integers '''A''' is a [[Bézout domain]].
| |
| | |
| ==References==
| |
| <references />
| |
| * Daniel A. Marcus, ''Number Fields'', third edition, Springer-Verlag, 1977
| |
| | |
| ==See also==
| |
| *[[Gaussian integer]]
| |
| *[[Eisenstein integer]]
| |
| *[[Root of unity]]
| |
| *[[Dirichlet's unit theorem]]
| |
| *[[fundamental unit (number theory)|Fundamental units]]
| |
| | |
| [[Category:Algebraic numbers]]
| |
| [[Category:Integers]]
| |
If you have a holiday home that you visit yourself or let loose for holidays, it is actually sat empty over a bitterly cold winter months when individuals are less in the climate for supplementing with break away in the united kingdom. This is a good chance to get within and obtain your maintenance done before the coldest weather arrives. Your property that isn't regularly occupied is inside a greater risk for various reasons than one that is inhabited full time, in which means you shouldn't neglect your holiday home. Indeed you should give it some extra care and attention. Check each room one by one and jot down any maintenance issues. Situations garden a transparent out too, ready for the spring. It could also be considered good time to give your next home completely new lick of paint.
Also, this best to be able to have trees and hedges that block the way to the doorway. If you a door without windows and hedges, and trees are blocking your windows, then the Qi doesn't place enter in. Try to at least trim the hedges, web templates of the window is showing and in the same time allowing for getting a bit of privacy. If privacy isn't an issue, trim the hedges into the bottom for this lower window sills.
This approach is used for hardwood plants. This is a type of pruning system. When crown thinning getting done, some stems and branches are selected and pruned. This kind of is done for being to boost your workers light penetration and air movement belonging to the crown for this tree. Bloodstream . the tree structure too.
A brush cutter is kind of like a miniature hand-held lawnmower that features a straight shaft and a head which has rotating blades - only of course, you can lift it for use well above the ground. Such a device will certainly take care of many pruning and restricting needs all of the garden, making the gardener's life a boatload easier. Such chores additional quickly done than should simply used hand-held tools like pruners or shears.
With all the above points, buying your home certainly sounds great! However, it is not for all of us. Every coin has 2 sides and home furniture think about every aspect before taking part in home possession. Of course, we don't need it to be our nightmare. When it comes to the purchase of a home, it really is a complicated, time-consuming and of course, costly endeavor. Why don't we think towards the responsibilities connected buying as opposed to. renting.
Lean Hog futures and options became liquid. Call and put options could be purchased cheaply at times, or particularly for fair prices. Lean hog futures often make good price moves. Live Hog cash margin requirements are concerning same as Live Cattle, about $1200. A one cent move equates to $400.
Did you know that another reliable tool is the Yard Message? Displaying a Available sign is strongly encouraged for advertising your home. It indicates a person are truly serious about selling your home.
For those who have virtually any questions regarding in which along with the way to use hedging, you can call us at our web-page.