Recommended Problems

1. (DF, 14.2, problem 14) Show that $\Q(\sqrt{2+\sqrt{2}})$ is an extension of degree $4$ with cyclic Galois group.

2. (DF, 14.2, Problem 16) Show that $x^4-2x^2-2$ is irreducible and that its roots are $\pm\sqrt{1\pm\sqrt{3}}$. Let $\alpha_{1}=\sqrt{1+\sqrt{3}}$ and $\alpha_{2}=\sqrt{1-\sqrt{3}}$. Show that $K_1=\Q(\alpha_1)$ and $K_2\Q(\alpha_2)$ are different, and that their intersection is the field $F=\Q(\sqrt{3})$. Then show that $K_1K_2$ has Galois group $\Z/2\Z\times\Z/2\Z$ over $F$. Finally show that $x^4-2x^2-2$ has galois group equal to the Dihedral group of the square.

3. (DF, 14.2, Problem 17-18) These problems derive some basic properties of the galois norm and trace for an algebraic element defined as:

• $\mathrm{Tr}(\alpha)=\sum_{\sigma}\sigma(\alpha)$ where the sum is over the set of Galois conjugates of $\alpha$
• $\mathrm{N}(\alpha)=\prod_{\sigma}\sigma(\alpha)$ where the product is over the set of Galois conjugates of $\alpha$

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