Achromatic lens

In mathematics, a polynomial is said to be irreducible if it cannot be factored into the product of two or more non-trivial polynomials whose coefficients are of a specified type. Thus in the common context of polynomials with rational coefficients, a polynomial is irreducible if it cannot be expressed as the product of two or more such polynomials, each of them having a lower degree than the original one. For example, while $x^{2}-1=(x-1)(x+1)$ is reducible over the rationals, $x^{2}+1$ is not.

For any field F, a polynomial with coefficients in F is said to be irreducible over F if it is non-constant and cannot be factored into the product of two or more non-constant polynomials with coefficients in F. The property of irreducibility depends on the field F; a polynomial may be irreducible over some fields but reducible over others. Some simple examples are discussed below.

A polynomial with integer coefficients, or, more generally, with coefficients in a unique factorization domain F is said to be irreducible over F if it is not invertible nor zero and cannot be factored into the product of two non-invertible polynomials with coefficients in F. This definition generalizes the definition given for the case of coefficients in a field, because, in this case, the non constant polynomials are exactly the polynomials that are non-invertible and non zero.

It is helpful to compare irreducible polynomials to prime numbers: prime numbers (together with the corresponding negative numbers of equal magnitude) are the irreducible integers. They exhibit many of the general properties of the concept of 'irreducibility' that equally apply to irreducible polynomials, such as the essentially unique factorization into prime or irreducible factors:

Every polynomial with coefficients in a field or a unique factorization domain F 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 invertible constants from F. This property of unique factorization is commonly expressed by saying that the polynomial rings over a field or a unique factorization domain are unique factorization domains. However the existence of such a factorization does not mean that, given a polynomial, the factorization may always be computed: there are fields such that it can not exist any algorithm to factorize polynomials over these fields. There exist factorization algorithms for the polynomials with coefficients in the rational numbers, in a finite field or a finitely generated field extension of these fields. They are described in the article Polynomial factorization.

If an univariate polynomial p has a root (in some field extension) which is also a root of an irreducible polynomial q, then p is a multiple of q, and thus all roots of q are roots of p; this is Abel's irreducibility theorem. This implies that the roots of an irreducible polynomial may not be distinguished through algebraic relations. This result is one of the starting points of Galois theory, which has been introduced by Évariste Galois to study the relationship between the roots of a polynomial.

Simple examples

The following six polynomials demonstrate some elementary properties of reducible and irreducible polynomials:

$p_{1}(x)=x^{2}+4x+4\,={(x+2)(x+2)}$,

$p_{2}(x)=x^{2}-4\,={(x-2)(x+2)}$,

$p_{3}(x)=9x^{2}-3\,=3(3x^{2}-1)\,=3(x{\sqrt {3}}-1)(x{\sqrt {3}}+1)$,

$p_{4}(x)=x^{2}-4/9\,=(x-2/3)(x+2/3)$,

$p_{5}(x)=x^{2}-2\,=(x-{\sqrt {2}})(x+{\sqrt {2}})$,

$p_{6}(x)=x^{2}+1\,={(x-i)(x+i)}$.

Over the ring $\mathbb {Z}$ of integers, the first three polynomials are reducible (the third one is reducible because the factor 3 is not invertible in the integers), the last two are irreducible. (The fourth, of course, is not a polynomial over the integers.)

Over the field $\mathbb {Q}$ of rational numbers, the first two and the fourth polynomials are reducible, but the other three polynomials are irreducible (as a polynomial over the rationals, 3 is a unit, and, therefore, does not count as a factor).

Over the field $\mathbb {R}$ of real numbers, the first five polynomials are reducible, but $p_{6}(x)$ is still irreducible.

Over the field $\mathbb {C}$ of complex numbers, all six polynomials are reducible. In fact, every nonzero polynomial $p(x)$ over $\mathbb {C}$ can be factored as

$p(x)=a(x-z_{1})\cdots (x-z_{n})$

where $n$ is the degree, $a$ the leading coefficient and $z_{1},\dots ,z_{n}$ the zeros of $p(x)$. Thus, the only non-constant irreducible polynomials over $\mathbb {C}$ are linear polynomials. This is the Fundamental theorem of algebra.

The existence of irreducible polynomials of degree greater than one (without zeros in the original field) historically motivated the extension of that original number field so that even these polynomials can be reduced into linear factors: from rational numbers ( $\mathbb {Q}$ ), to the real subset of the algebraic numbers ( ${\mathcal {A}}\cap \mathbb {R}$ ), and finally to the algebraic subset of the complex numbers ( ${\mathcal {A}}\cap \mathbb {C}$ ). After the invention of calculus those latter two subsets were later extended to all real numbers ( $\mathbb {R}$ ) and all complex numbers ( $\mathbb {C}$ ).

For algebraic purposes, the extension from rational numbers to real numbers is too "radical": it introduces transcendental numbers, which are not the solutions of algebraic equations with rational coefficients. These numbers are not needed for the algebraic purpose of factorizing polynomials (but they are necessary for the use of real numbers in analysis). The set of algebraic numbers ( ${\mathcal {A}}$ ) is the algebraic closure of the rationals, and contains the roots of all polynomials (including i for instance). This is a countable field and is strictly contained in the complex numbers – the difference being that this field ( ${\mathcal {A}}$ ) is "algebraically complete" (as are the complex numbers, $\mathbb {C}$ ) but not analytically complete since it lacks the aforementioned transcendentals.

The above paragraph generalizes in that there is a purely algebraic process to extend a given field F with a given polynomial $p(x)$ to a larger field where this polynomial $p(x)$ can be reduced into linear factors. The study of such extensions is the starting point of Galois theory.

Real and complex numbers

As shown in the examples above, only linear polynomials are irreducible over the field of complex numbers (this is a consequence of the fundamental theorem of algebra). Since the complex roots of a real polynomial are in conjugate pairs, the irreducible polynomials over the field of real numbers are the linear polynomials and the quadratic polynomials with no real roots. For example, $x^{4}+1$ factors over the real numbers as $(x^{2}+{\sqrt {2}}x+1)(x^{2}-{\sqrt {2}}x+1).$

Generalization

If R is an integral domain, an element f of R which is neither zero nor a unit is called irreducible if there are no non-units g and h with f = gh. One can show that every prime element is irreducible; the converse is not true in general but holds in unique factorization domains. The polynomial ring F[x] over a field F (or any unique-factorization domain) is again a unique factorization domain. Inductively, this means that the polynomial ring in n indeterminants (over a ring R) is a unique factorization domain if the same is true for R.

Finite fields

Factorization over a finite field behaves similarly to factorization over the rational or the complex field. However, polynomials with integer coefficients that are irreducible over the field $\mathbb {Q}$ can be reducible over a finite field. For example, the polynomial $x^{2}+1$ is irreducible over $\mathbb {Q}$ but reducible over the field $\mathbb {F} _{2}$ of two elements. Indeed, over $\mathbb {F} _{2}$, we have

$(x^{2}+1)=(x+1)^{2}$

The irreducibility of a polynomial over the integers $\mathbb {Z}$ is related to that over the field $\mathbb {F} _{p}$ of $p$ elements (for a prime $p$). Namely, if a polynomial over $\mathbb {Z}$ with leading coefficient $1$ is reducible over $\mathbb {Z}$ then it is reducible over $\mathbb {F} _{p}$ for any prime $p$. The converse, however, is not true, there are polynomials of arbitrary large degree that are irreducible over the integers and reducible over every finite field. A simple example of such a polynomial is $x^{4}+1,$ which is irreducible over the integers and reducible over every finite field.

See also

• Gauss's lemma (polynomial)
• Rational root theorem, a method of finding whether a polynomial has a linear factor with rational coefficients
• Eisenstein's criterion
• Perron method
• Hilbert's irreducibility theorem
• Cohn's irreducibility criterion
• Irreducible component of a topological space
• Factorization of polynomials over finite fields
• Quartic function#Factorization into quadratics
• Cubic function#Factorization
• Casus irreducibilis, the irreducible cubic with three real roots
• Quadratic equation#Quadratic factorization

References

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• 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, pp. 91.

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Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.