Answer to Question #215693 in Abstract Algebra for Wavie

Question

4. Find Φ12(x) over Q.

Answer 1

Solution

We know that there are $\phi$ (12) = 4 primitive 12th roots of unity. If

ω is one such a root, the others are $\omega^5$ , $\omega^7$ and $\omega^{11}$. We know that

(x¹²− 1) = (x⁶ − 1)(x⁶+ 1) .

$\phi$ (12) is a factor of (x6 + 1).

Furthermore, we know that

i and -i are non-primitive roots of unity, so we can remove a factor of x² + 1, to get x⁴-x²+1.

Since it is a polynomial of 4 degree

It's the required one.

Answer:

$\phi_{12}(x)=x^4-x^2+1$