Recommended Problems

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

(DF, 14.2, Problem 16) Show that $x^42x^22$ 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 $x^42x^22$ has galois group equal to the Dihedral group of the square.

(DF, 14.2, Problem 1718) 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 $\alpha$