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Abstract Algebra

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Moderate

Splitting field of $K/F$: Definition and Examples

ABSALG-MH2NPW

By the Fundamental Theorem of Algebra, every polynomial $f(x)\in {\mathbb C}[x]$ is a product of linear factors in ${\mathbb C}[x]$.

Let $F\subseteq{\mathbb C}$ be a field. If $f(x)\in F[x]$, and $f(x)=a_n(x-\alpha_1)\ldots(x-\alpha_n)$, $\alpha_1,\ldots,\alpha_n\in{\mathbb C}$, then the splitting field over $F$ of $f$ is the field $F(\alpha_1,\ldots,\alpha_n)$ generated over $F$ by all the roots of $f(x)$.

Which of the following are true?

Select ALL that apply.

A

The splitting field over ${\mathbb Q}$ of the polynomial $x^2-2$ is ${\mathbb Q}(\sqrt{2})$

B

The splitting field over ${\mathbb Q}$ of $x^3-2$ is ${\mathbb Q}(\sqrt{-1},\sqrt[3]{2})$, $\sqrt[3]{2}\in{\mathbb R}$

C

There is a prime $p\ge 2$ such that the splitting field over ${\mathbb Q}$ of $x^p-1$, $p$ prime, has degree $p$ over ${\mathbb Q}$

D

The splitting field over ${\mathbb Q}$ of $x^3-2$ is ${\mathbb Q}(\omega, \sqrt[3]{2})$, $\sqrt[3]{2}\in{\mathbb R}$ where $\omega^2+\omega+1=0$

E

There is a prime $p\ge 2$ such that the splitting field over ${\mathbb Q}$ of $x^p-2$, $p\ge2$ prime, has degree $p^2$