Eisenstein's Criterion - Examples

Examples

Consider the polynomial Q = 3x4 + 15x2 + 10. In order for Eisenstein's criterion to apply for a prime number it must divide both non-leading coefficients 15 and 10, which means only could work, and indeed it does since 5 does not divide the leading coefficient 3, and its square 25 does not divide the constant coefficient 10. One may therefore conclude that Q is irreducible over the rational numbers (and since it is primitive, over the integers as well). Note that since Q is of degree 4, this conclusion could not have been established by only checking that Q has no rational roots (which eliminates possible factors of degree 1), since a decomposition into two quadratic factors could also be possible.

Often Eisenstein's criterion does not apply for any prime number. It may however be that it applies (for some prime number) to the polynomial obtained after substitution (for some integer a) of x + a for x; the fact that the polynomial after substitution is irreducible then allows concluding that the original polynomial is as well. This procedure is known as applying a shift.

For example consider H = x2 + x + 2, in which the coefficient 1 of x is not divisible by any prime, Eisenstein's criterion does not apply to H. But if one substitutes y + 3 for x in H, one obtains the polynomial x2 + 7x + 14, which satisfies Eisenstein's criterion for the prime number 7. Since the substitution is an automorphism of the ring, the fact that we obtain an irreducible polynomial after substitution implies that we had an irreducible polynomial originally. In this particular example it would have been simpler to argue that H (being monic of degree 2) could only be reducible if it had an integer root, which it obviously does not; however the general principle of trying substitutions in order to make Eisenstein's criterion apply is a useful way to broaden its scope.

Another possibility to transform a polynomial so as to satisfy the criterion, which may be combined with applying a shift, is reversing the order of its coefficients, provided its constant term is nonzero (without which it would be divisible by x anyway). This is so because such polynomials are reducible in R if and only if they are reducible in R (for any integral domain R), and in that ring the substitution of x−1 for x reverses the order of the coefficients (in a manner symmetric about the constant coefficient, but a following shift in the exponent amounts to multiplication by a unit). As an example 2x5 − 4x2 − 3 satisfies the criterion for p = 2 after reversing its coefficients, and (being primitive) is therefore irreducible in ℤ.