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\section*{Math 403/503 Spring 2024}
\subsection*{Homework 9, due April 17}
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\item Let $E/F$ be a field extension. If $\alpha,\beta,\gamma\in E$ are algebraic over $F$, explain why $\alpha+\beta+\gamma$ is algebraic over $F$.
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\item Let $E$ be a splitting field for an irreducible polynomial $p(x)\in F[x]$ over $F$, and $\alpha$ and $\beta$ be two roots of $p(x)$. Show that there is an isomorphism $\Phi:E\to E$ such that $\Phi(\alpha)=\beta$.
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\item Prove that the fields $\Q(\sqrt[4]{3})$ and $\Q(\sqrt[4]{3}i)$ are isomorphic but not equal.
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\item Prove or disprove: $\Q(\sqrt{2})\cong\Q(\sqrt{3})$.
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\item Let $K$ be an algebraic extension of $E$, and $E$ an algebraic extension of $F$. Prove that $K$ is algebraic over $F$. [Caution: Do not assume that the extensions are finite.]
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