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설대 셤

Q : Describe the Galois group of the polynomial $x^5-3\in\Bbb Q(\eta)[x]$ over $\Bbb Q(\eta)$ where $\eta$ is a primitive $5$th root of unity.\\

A : The splitting field of $x^5-3$ over $\Bbb Q(\eta)$ is $E:=\Bbb Q(\eta,\sqrt[5]{3})$ with degree $[E:\Bbb Q(\eta)] = 5$ as $[E:\Bbb Q]\mid 5$ and $[E:\Bbb Q]\mid \varphi(5) = 4$. Hence, $Gal(E/\Bbb Q(\eta)) =\Bbb Z/5$ which is cyclic.\\

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카이 셤

Q : Represent Galois group of the polynomial $x^{33}-3\in\Bbb Q[x]$ as a semidirect product and compute its degree.\\

A : The splitting field of $x^{33}-3\in\Bbb Q[x]$ is $E:=\Bbb Q(\sqrt[33]{3},\eta)$ where $\eta$ is a primitive $33$th root of unity. Now, $[E:\Bbb Q]\mid\varphi(33) = 20, [E:\Bbb Q]\mid 33$. Since $\gcd(33,20) =1$, we conclude that $[E:\Bbb Q] = 33\cdot 20 = 660$. Note that $\Bbb Q(\eta)/\Bbb Q$ is a Galois, $N = Gal(E/\Bbb Q(\eta))$ is normal in $Gal(E/\Bbb Q)$. Hence we have a s.e.s.

$$0\to N\to G\to G/N\to 0$$

Here, $G/N\simeq Gal(\Bbb Q(\eta)/\Bbb Q)$ and as $Gal(E/\Bbb Q(\sqrt[33]{3}))\simeq Gal(\Bbb Q(\eta)/\Bbb Q)$, the sequence splits. Hence, $G\simeq N\rtimes(G/N)$.

설대 갈루아를 주는 문제로 출제하다니.. 맘에 안든다.