Eisenstein's criterion

The Eisenstein criterion or Irreduzibilitätskriterium Eisenstein used in algebra to prove the irreducibility of a given polynomial. It can thus be easier statements about the divisibility of polynomials meet.

The criterion is named after the mathematician Gotthold Eisenstein, who in 1850 to a high-profile article in Crelle Journal ( Volume 39 ) wrote. Just four years earlier, it had been ibid. first published by Theodor Beautiful Man ( Volume 32 ). It was and is partially named after Schönemann.

The criterion

Be a polynomial with integer coefficients, ie

If a prime number exists, to notify all coefficients, the coefficients but not square and do not share; if so

• For all and
• And

Holds, then is irreducible.

Generalization

If the coefficients of a factorial ring and there is a corresponding prime element, so the polynomial is irreducible in the polynomial ring of the quotient field of

Comments

• A polynomial for which it exists, is also called Eisenstein polynomial with respect.
• The criterion is only sufficient; even if it is not satisfied, the polynomial may be irreducible. The dismantling of a polynomial can therefore not be detected.
• For a decomposition into one can use the criterion as follows. Of course, it is true: has content 1 and is irreducible to conclude in irreducible in Holds allowed so as to Diophantine equation for x, then: If the criterion is met for, so there is no integer solution of the equation.
• However, from the Gauss lemma follows the converse: in irreducible irreducible in

Examples

• Is according to the above criterion ( choose ) irreducible over. This means that the real root of the polynomial must be irrational.
• Is irreducible in if is a prime number or a simple divisors has. In particular, can then be rational for no.
• Not fulfill the criterion and is irreducible. meets the criterion as little, but can be disassembled into
• Meets the criterion with, so is irreducible in the polynomial paths but reducible in, because it splits into two non- units.

Evidence

The proof is by contradiction: Suppose a Eisensteinpolynom would respect and there are two non-constant polynomials and since, by assumption, all but the leading coefficient divisible by, modulo the following argument applies: This must also be modulo monomials and, d h their other coefficients are all divisible by. In particular, the constant terms of, and are each divisible by. Because follows with the Cauchy product that the constant term of is divisible by - Opposition to the fact that the criterion for satisfied. This must be irreducible, and that was just to show.

Considering general polynomials over a factorial ring, so the modulo argument must be replaced by a suitable homomorphism, which in its corresponding residue class maps. There is factorial and a prime element, can be the homomorphism find them easily. The linearity allows analogy then the conclusion that each and ready itself to a monomial be.