example of using Eisenstein criterion

For showing the irreducibility (http://planetmath.org/IrreduciblePolynomial2) of the polynomialPlanetmathPlanetmath


one would need a prime numberMathworldPlanetmath dividing its other coefficients except the first one, but there is no such prime.  However, a suitable  x:=y+a  may change the situation.  Since  the binomial coefficientsMathworldPlanetmath of (y-1)5 except the first and the last one are divisible by 5 and 111(mod5),  we try




Thus the prime 5 divides other coefficients except the first one and the square of 5 does not divide the constant term of this polynomial in y, whence the Eisenstein criterion says that P(y-1) is irreducible (in the field of its coefficients).  Apparently, also P(x) must be irreducible.

It would be easy also to see that P(x) does not have rational zeroes (http://planetmath.org/RationalRootTheorem).

Title example of using Eisenstein criterion
Canonical name ExampleOfUsingEisensteinCriterion
Date of creation 2013-03-22 19:10:14
Last modified on 2013-03-22 19:10:14
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 5
Author pahio (2872)
Entry type Example
Classification msc 13A05
Classification msc 11C08