Answer to Question #271307 in Abstract Algebra for N. Kokom

Question

Determine which of the polynomials below is (are) irreducible over Q. a. x5 1 9x4 1 12x2 1 6 b. x4 1 x 1 1

Answer 1

a.

Eisenstein's criterion:

If there exists a prime number p such that the following three conditions all apply:

• p divides each a_i for 0 ≤ i < n,
• p does not divide a_n,
• p² does not divide a₀,

then polynomial is irreducible over the rational numbers.

we have:

3 divides each a_i for 0 ≤ i < n: $9/3=3,12/3=4,6/3=2$

3 does not divide a₅ =1

3²=9 does not divide a₅=1

$x^5 + 9x^4 + 12x^2 + 6$

is irreducible according Eisenstein’s criterion with p = 3:

b.

$x^4 + x + 1$

Consider $x^4 + x + 1$ mod 2. It is easy to see that this polynomial has no roots in Z₂, and so to prove irreducibility in Z₂ it suffices to show it has no quadratic factors. The only quadratic polynomial in Z₂[x] that does not have a root in Z₂ is $x^2 + x + 1$ which does not divide $x^4 + x + 1$ in Z₂[x], as is also easily checked. It follows that $x^4 + x + 1$ is irreducible in Z₂[x] and so by the mod p test with p = 2 we conclude that $x^4 + x + 1$ is irreducible in Q[x].