example of using Eisenstein criterion
For showing the irreducibility (http://planetmath.org/IrreduciblePolynomial2) of the polynomial
one would need a prime number dividing its other coefficients except the first one, but there is no such prime. However, a suitable may change the situation. Since the binomial coefficients of except the first and the last one are divisible by 5 and , 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 , whence the Eisenstein criterion says that is irreducible (in the field of its coefficients). Apparently, also must be irreducible.
It would be easy also to see that does not have rational zeroes (http://planetmath.org/RationalRootTheorem).
|Title||example of using Eisenstein criterion|
|Date of creation||2013-03-22 19:10:14|
|Last modified on||2013-03-22 19:10:14|
|Last modified by||pahio (2872)|