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3 votes
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What is the fixed field of $\mathbb{Q}(\sqrt[3]{5}, \sqrt[3]{5}\zeta_3)$ with 3-cycle group?

The fixed field is $\mathbb{Q}(\zeta_3)$. To verify that $\zeta_3$ lies in the fixed field, note that $$\zeta_3 = \frac{\sqrt[3]{5}\zeta_3}{\sqrt[3]{5}}.$$ So the image of $\zeta_3$ under the ...
Arturo Magidin's user avatar
2 votes

$K_1,K_2$ are isomorphic subfields of $L$ via $\sigma$, every polynomial $P(x)$ over $K_1$ have the same number of roots in $L$ as $\sigma(P(x))$?

The problem is that $K_1$, $K_2$ may be isomorphic, but not "sitting the same way" in $L$. Now, an example, $K_1= L = k(t)$, and $K_2 = k(t^2)$, with the isomorphism $\sigma \colon t\mapsto ...
orangeskid's user avatar
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4 votes
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$K_1,K_2$ are isomorphic subfields of $L$ via $\sigma$, every polynomial $P(x)$ over $K_1$ have the same number of roots in $L$ as $\sigma(P(x))$?

No, not always. Let $K_1=K_2=\Bbb{Q}(\sqrt2)$, $\sigma(\sqrt2)=-\sqrt2$. Let $L=\Bbb{Q}(\root4\of2)\subseteq\Bbb{R}$. Then $P(x)=x^2-\sqrt2$ has two roots in $L$ whereas $\sigma(P(x))=x^2+\sqrt2$ has ...
Jyrki Lahtonen's user avatar

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