Fields4

From Exampleproblems

Show that the splitting field of $f(x)=x^4+1\in\mathbb{Q}[x]$ is a simple extension of $\mathbb{Q}$ .

Let $\zeta_8=\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i$ be a primitive 8th root of unity and note that $\zeta_8\,$ is a root of $f(x)=x^4+1\,$ . (Make sure you understand why this is true.) Further, as $\zeta_8^8=1$ and $\zeta_8^4=-1$ , note that $\zeta_8^3,\zeta_8^5,\zeta_8^7$ are all roots of $f(x)\,$ as well. Thus, we have found the four roots of this quartic polynomial, so the splitting field of $f\,$ is $E=\mathbb{Q}(\zeta_8,\zeta_8^3,\zeta_8^5,\zeta_8^7)$ . As $E\,$ is a field, all the integral powers of $\zeta_8\,$ must be elements of it, so we may write $E=\mathbb{Q}(\zeta_8)$ , showing that $E\,$ is a simple extension of $\mathbb{Q}$ .