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This answer is meant to answer only your second question. Claim. Let $K=\mathbb{Q}(\sqrt[3]{2})$ and $\alpha = \sqrt[3]{2}-\sqrt[3]{4} \in K$. Then there does not exist $\beta \in K$ such that $\beta^... • 4,380 1 vote ### Algebraic numbers which prescribed degree which does not belong to some fields Proposition 2 is false, although perhaps only for$n = 4$(and$t = 2$or$t = 3$). If$t = 2$, every algebraic number$\gamma$of degree$4$is contained in$K_{4}$. Let$K$be the Galois closure of$...
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Proposition 1 is true. Let $x$ be an element of degree $n$ whose Galois group is isomorphic to $S_n$. Let $F$ be a finite subset of algebraic elements of degree $<n$. I claim that $x$ is not in the ...