Tutorial 7 : Galois Groups of Quartics and Solvability by Radicals

1. Determine the Galois groups of the quartics: $ x^4-2, \;\; x^4+2,\;\; x^4-x+1,\;\; x^4+x+1, x^4+x^3+x^2+x+1,\;\;$   and$ \;\; x^4+4x^2-5.$

2. Show that the resolvent cubic $ r(x)$ of $ f(x)=x^4+1$ is $ x(x-2)(x+2)$ and $ G_f=V.$

3. Show that the Galois group of an irreducible quartic in $ \mathbb{Q}[x]$ with exactly two real roots is either $ S_4$ or $ D_4.$

4. Let $ \alpha$ be a real root of an irreducible rational quartic whose resolvent cubic is irreducible. Show that $ \alpha$ is not constructible by ruler and compass. Can we construct the roots of the quartic $ x^4+x-5$ by ruler and compass ?

5. Show that $ f(x)=x^4-2x^2-1$ is irreducible over $ \mathbb{Q}$ and $ G_f=D_4.$

6. Find sufficient conditions on the integers $ a,b$ and $ c$ so that $ \mathbb{Q}(\sqrt{a+b\sqrt{c}})$ is a Galois extension of $ \mathbb{Q}$ with cyclic Galois group of order $ 4.$

7. Let $ f(x) \in
\mathbb{Q}[x]$ be an irreducible polynomial of prime degree $ p \geq
5$ Suppose that $ f(x)$ has exactly two non-real roots. Show that $ f(x)$ is not solvable by radicals over $ \mathbb{Q}.$

8. Let $ x_1,x_2,x_3$ be indeterminates and let $ s_1, s_2, s_3$ be the elementary symmetric polynomials of $ x_1,x_2,x_3.$ Show that $ E=\mathbb{Q}(x_1,x_2,x_3)$ is not a radical extension of $ F=\mathbb{Q}(s_1,s_2,s_3)$ but $ \mathbb{Q}(\zeta_3)(x_1,x_2,x_3)$ is a radical extension of $ \mathbb{Q}(\zeta_3, s_1,s_2,s_3).$

9. Let $ G$ be the Galois group of an irreducible quintic over $ \mathbb{Q}.$ Show that $ G=A_5$ or $ S_5$ if $ G$ has an element of order $ 3.$