Irreducible polynomial

In algebra, a branch of mathematics, an irreducible polynomial is a polynomial that can not be written as a product of two non- invertible polynomials and thus not divided into " simple " polynomials. Their importance for the polynomial is in most cases ( polynomials over factorial rings ) with the meaning of prime numbers are natural numbers equal.

  • 3.1 The Irreduzibilitätskriterium Eisenstein
  • 3.2 Reduction criterion

Definition

The definition can already formulate integrity rings. It is known that the polynomial itself is zero divisors over an integral domain. This is the reason that the definitions can be given of irreducible elements. Since in many cases only the body to be treated and there is the definition easier, even the definition of this special case is indicated. In the general definition can be limited to a variable itself trivially.

General definition for integrity Rings

It should be an integral domain. Then a polynomial is called irreducible, if not invertible in and is for either or invertible.

Definition specifically for body

It is a body. Then is called a polynomial in the polynomial ring in indeterminates irreducible if is not constant and there are no non-constant polynomials, so that is true. If such polynomials exist, is also called reducible or decomposable.

An equivalent description is: Irreducible polynomials are exactly the irreducible elements in the ring.

Prime polynomials and irreducible polynomials in comparison

A polynomial is called prime, or prime polynomial, if follows with the property or for all. If the ring even factorial, so is factorial ( Gauss ). In particular, all bodies are factorial and thus the associated polynomial.

For polynomials over factorial rings (thus also for polynomials over a field ) are prime elements and irreducible elements and vice versa. It also applies a unique up to Associated awareness decomposition of polynomials in prime polynomials.

It can be in these factorial rings, the irreducibility of polynomials also due to the irreducibility of polynomials over the quotient field. This problem is not necessarily easier to solve. Note to the fact that a polynomial of a factorial ring is exactly then prime if the polynomial constant either a prime or irreducible and primitive (ie, the greatest common divisor of all coefficients ) in the quotient field above.

Irreduzibilitätskriterien

In many areas polynomials occur in a variable whose irreducibility makes further inferences possible, for example, fundamentally in the Galois theory and application examples as the chromatic polynomial in graph theory. ( See also minimal polynomial ). It is therefore important to have easy decision criteria for the irreducibility at hand.

The Irreduzibilitätskriterium Eisenstein

The Eisenstein criterion is a sufficient criterion for the irreducibility of a polynomial in an extended set of coefficients. For this purpose let an integral domain, a polynomial with coefficients in the quotient and body of. If one finds a prime element, such that:

  • For and

Then is irreducible over. It is often applied for and. It is the condition of divisibility by the prime element everywhere replaced by containment in a prime ideal of.

Is factorial and polynomial primitive, ie. is the greatest common divisor of all the coefficients, then is also irreducible.

Reduction criterion

It is again an integral domain with quotient field and a prime element. A polynomial is then (not necessarily accurate then ) in irreducible if the polynomial with coefficients in reduced modulo irreducible.

Examples

  • About bodies shall: Every polynomial of degree 1 is irreducible. If an irreducible polynomial is a zero, so it has degree 1
  • In particular, each irreducible polynomial has over an algebraically closed field such as degree 1
  • Every polynomial over of degree 2 or degree 3 is irreducible if and only if it has no zeros in.
  • Each irreducible polynomial over the real numbers is 1 or 2 degrees, therefore, either the shape or. This is due to the fact that the algebraic closure degree 2 has.
  • Over irreducible over for a prime number from, or is primitive and irreducible
  • Is irreducible. Because the polynomial is invariant under the induced picture, it would otherwise decompose into linear factors, but this can not be, since the polynomial has no zeros in.
  • The polynomial is irreducible, because it is primitive and an irreducible polynomial in the rational numbers. We apply this to the reduction criterion. The polynomial with coefficients modulo is reduced, and this is irreducible.
  • Is irreducible. This follows from the Eisenstein criterion with the prime element.
  • For a prime number, the polynomial is, irreducible over. The minimal polynomial of over is so. As a corollary arises, for example, that the square root of an irrational number (or a - th root of a prime number ).
  • (or as an element of - note that it is primitive. ) is irreducible (iron Steinsches criterion). The prime element is. However, this polynomial is separable, ie. it has the algebraic degree of a multiple zero. This phenomenon does not occur in.
Unique factorization domain Content (algebra) Primitive polynomial (field theory) Minimal polynomial (linear algebra)
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