Prove that $x^{6}+30x^{5}-15x^{3}+6x-120$ can't be written as a product of two polynomials of rational coefficients and positive degrees.
$x^{6}+30x^{5}-15x^{3}+6x-120$can't be written as products of two polynomials of rational coefficients and positive degrees.
1
$\begingroup$
number-theory
polynomials
-
0Eisenstein polynomial. – 2012-12-12
-
0In that case we say the polynomial is **irreducible** over the complex numbers. That will give you something to look up. – 2012-12-12
-
0@GEdgar over the complex numbers? – 2012-12-12
-
2Look at $p=3$ and use Eisenstein. – 2012-12-12
-
0The question says rational coefficients. If the OP does not use the word "irreducible" then he/she likely does not know what "Eisenstein" means. – 2012-12-12
1 Answers
4
Suppose $f(x) = x^{6}+30x^{5}-15x^{3}+6x-120$ is not irreducible in $\mathbb{Q}[x]$. Then it is not irreducible in $\mathbb{Z}[x]$ by Gauss's lemma. However it is irreducible in $\mathbb{Z}[x]$ by Eisenstein's criterion using the prime number $3$. This is a contradiction. Hence $f(x)$ is irreducible in $\mathbb{Q}[x]$