TUT ALG3 8

Problem 1

Exercise 346.1(Exercise).

Let $F_1\subseteq F_2\subseteq F_3$ be fields. Show that $[F_3:F_1]=[F_3:F_2][F_2:F_1]$.

Let $[F_3:F_2]=p$ and $[F_2:F_1]=q$.

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Problem 2

[!Exercise]
Is $\pi \cdot i$ algebraic over $\mathbb{R}$? Is it algebraic over $\mathbb{Q}$?

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Problem 3

[!Exercise]
Show that $\sqrt{2}+\sqrt{3}$ is algebraic over $\mathbb{Q}$. Compute

${\alpha }^4-10{\alpha }^2+1$.

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Problem 4