Prove that the polynomials x4 x3 x2 x 1 and x4 x3 x2

Prove that the polynomials x^4 + x^3 + x^2 + x + 1 and x^4 - x^3 + x^2 - x + 1 in Q[x] are irreducible.

Solution

Consider, f₄[X]= x⁴ + x³ + x² + x + 1

So, f₀ = 1 , which is not equal to 0

Now, f1 = 1⁴ + 1³ + 1² + 1 + 1 = 1 , which is not equal to 0

Also, f2 = 2⁴ + 2³ + 2² + 2 + 1 = 3, which is not equal to 0

Lastly, f3 = 3⁴ + 3³ + 3² + 3 + 1 = 1, which is not equal to 0

Thus,  f is irreducible over Fp[X] then f is irreducible over Z[X]..

Hence, x⁴ + x³ + x² + x + 1 is irreducible in Q{x}.

Similarly, x⁴ + x³ + x² - x + 1 is irreducible in Q{x}.