Let the field extension $\mathbb{Q} \leq \mathbb{C}$ and $a=e^{\frac{2 \pi i}{8}}$. I have to find the minimal irreducible polynomial $Irr(a, \mathbb{Q})$.
$a$ is a root of the polynomial $f(x)=x^8-1=(x^4-1)(x^4+1)$.
$x^4-1$ is not irreducible, since $x^4-1=(x^2-1)(x^2+1)$.
So, we have to determine if the polynomial $x^4+1$ is irreducible, right??
How could I do that??
Two ways:
Suppose $x^4 + 1$ is reducible. Then it is the product of two polynomials of smaller degree. The possibilities for these degrees are $1$, $3$ or $2$, $2$. Show that $x^4 + 1$ cannot have a rational root, and deduce that the case $1$, $3$ is not possible. Can $a$ be the root of a quadratic over the rationals?
Alternatively, apply the Eisenstein criterion to $(x+1)^4 + 1$.