Chapter 12: Galois Theory

Construct a polynomial in $\mathbf{\mathbb{Q}}\mathbf{\left[}\mathbf{x}\mathbf{\right]}$ of degree 7 whose Galois group is S₇.

Show that ${\mathbf{x}}^{\mathbf{2}}\mathbf{-}\mathbf{3}$ and ${\mathbf{x}}^{\mathbf{2}}\mathbf{-}\mathbf{2}\mathbf{x}\mathbf{-}\mathbf{2}\mathbf{\in }\mathbf{Q}\mathbf{\left[}\mathbf{x}\mathbf{\right]}$ have the same Galois group.

Assume $\mathit{[}\mathit{K}\mathit{:}\mathit{F}\mathit{]}$ is finite. Is it true that every F-automorphism of Kis completely determined by its action on a basis of Kover F?

Let K is a normal extension of Q and $\left[K:Q\right]\mathbf{=}\mathit{P}$ with p prime show that $\mathit{G}\mathit{a}{\mathit{l}}_{\mathbf{Q}}\mathit{K}\mathbf{\cong }{\mathit{Z}}_{\mathbf{P}}$.

Question: (a) show that $\mathit{\omega }\mathbf{=}\left(-1+\sqrt{3}i\right)\mathbf{/}\mathbf{2}$is a root of${\mathit{x}}^{\mathbf{3}}\mathbf{-}\mathbf{1}$.

(b) Show that$\mathbf{\omega }$and${\mathbf{\omega }}^{\mathbf{2}}$are the root of${\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathit{x}\mathbf{+}\mathbf{1}$. Hence$\mathit{Q}\left(\omega \right)$is the splitting field of${\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathit{x}\mathbf{+}\mathbf{1}$.

If K is radical extension of F prove that $\mathbf{[}\mathbf{K}\mathbf{:}\mathbf{F}\mathbf{]}\mathbf{}$is finite.

If$\left[K:F\right]$ is finite, $\mathit{\sigma }\mathbf{\in }\mathit{G}\mathit{a}{\mathit{l}}_{\mathbf{F}}$, and$\mathit{u}\mathbf{\in }\mathit{K}$ is such that $\mathit{\sigma }\left(u\right)$, show that $\mathit{\sigma }\mathbf{\in }\mathit{G}\mathit{a}{\mathit{l}}_{\mathbf{F}}$.

Prove that for $\mathit{n}\mathbf{\ge }\mathbf{5}\mathbf{,}{\mathit{A}}_{\mathbf{n}}$ is not solvable.

Write out the operation table for the group $\mathit{G}\mathit{a}{\mathit{l}}_{\mathbf{Q}}\mathbf{}\mathbf{(}\sqrt{\mathbf{3}}\mathbf{,}\sqrt{\mathbf{5}}\mathbf{)}\mathbf{=}\left\{i,\tau .\alpha ,\beta \right\}$.

Let$\mathit{f}\left(x\right)\mathbf{\in }\mathit{F}\left[x\right]$ be separable of degreerole="math" localid="1657966853457" $\mathit{n}$ and$\mathit{k}$ a splitting field of$\mathit{f}\left(x\right)$ . Show that the order of$\mathit{G}\mathit{a}{\mathit{l}}_{\mathbf{F}}\mathit{k}$ divides$\mathit{n}\mathbf{!}$ .