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Abstract Algebra

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Splitting Field: Minimal Polynomial Prime Root, Relative Degree

ABSALG-F8LCZM

Let $f(x)=x^p-5$, where $p\ge 3$ is prime, and let $\alpha=\sqrt[p]{5}\in\mathbb{R}$ be the real $p$-th root of $5$.

Let $\zeta_p=\exp(2\pi i/p)$.

Let $K$ be the splitting field of $f(x)$ over $\mathbb{Q}$.

Let $\Phi_p(x)=(x^p-1)/(x-1)=x^{p-1}+x^{p-2}+\ldots+x^2+1\in\mathbb{Q}[x]$ be the $p$-th cyclotomic polynomial.

Which of the following are true?

Select ALL that apply.

A

$[K:\mathbb{Q}(\zeta_p)]=p-1$

B

$[K:\mathbb{Q}(\zeta_p)]=p$

C

$[K:\mathbb{Q}(\alpha)]=p-1$

D

$[K:\mathbb{Q}(\alpha)]=p$

E

$f(x)$ is irreducible over $\mathbb{Q}(\zeta_p)$.

F

$\Phi_p(x)$ is reducible over $\mathbb{Q}(\alpha)$.