Answer to Question #133624 in Abstract Algebra for PRATHIBHA ROSE C S

Question

Check whether x^5+9x^4+12x^2+6 is reducible over Q.

Answer 1

Given polynomial is $f(x) = x^5+9x^4+12x^2+6$ .

Compare the polynomial with $f(x) = a_5 x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0$ , we have

$a_5 = 1, a_4= 9, a_3 = 0, a_2 = 12, a_1 = 0, a_0 = 6$ .

There esist a prime number $p= 3$ which divides all coefficients $a_4,a_3,a_2,a_1,a_0$ , but not the leading coefficient $a_5$ and, $p^2$ not divide the constant term .

Hence, using the Einstein Criteria, given polynomial is irreducible over $Q$ .