Tutorial 5 : Fundamental Theorem of Galois Theory


1. Let $ f(x)\in \mathbb{Z}[x]$ be an irreducible quartic with Galois group $ S_4$ over $ \mathbb{Q}.$ Let $ \theta$ be a root of $ f(x).$ Show that there is no field properly contained in $ \mathbb{Q}(\theta)/\mathbb{Q}.$ Is $ \mathbb{Q}(\theta)/\mathbb{Q}$ a Galois extension ?



2. Consider the biquadratic extension $ E = \mathbb{Q}(i,\sqrt{2})$ of $ \mathbb{Q}.$ Find the intermediate subfields of $ E/\mathbb{Q}$ and match them with the subgroups of the Galois group $ G(E/\mathbb{Q}).$



3. Let $ E=\mathbb{C}(t)$ where $ t$ is a transcendental over $ \mathbb{C}.$ Let $ \omega= e^{2\pi i/3}.$ Define the $ \mathbb{C}$-automorphisms $ \sigma$ and $ \tau$ of $ E$ by the equations $ \sigma(t)=\omega t$ and $ \tau(t)= 1/t.$ Show that
$\displaystyle \sigma^3=\tau^2=id \;\;$and $\displaystyle \;\;
\tau \sigma =\sigma^{-1} \tau.$
Show that the group $ G$ of automorphisms generated by $ \sigma$ and $ \tau$ has order $ 6$ and $ E^G=\mathbb{C}(t^3+t^{-3}).$



4. Consider the polynomial $ f(x)= x^4-2x^2-2.$

      (a)
Show that the roots of the quartic are
$\displaystyle \alpha_1= \sqrt{1+ \sqrt{3}}, \,
\alpha_2= \sqrt{1- \sqrt{3}}, \,
\alpha_3= -\alpha_1 \,$ and $\displaystyle \;\alpha_4= -\alpha_2.$
      (b) Show : $ K_1=\mathbb{Q}(\alpha_1) \neq K_2=\mathbb{Q}(\alpha_2)$ and $ K_1 \cap K_2=\mathbb{Q}(\sqrt{3})= F.$
      (c)
Show that $ K_1,$ $ K_2$ and $ K_1K_2$ are Galois over $ F$
      (d)
Show that $ G(K_1K_2/F)$ is the Klein $ 4$-group. Determine the automorphisms in             this group.
      (e) Show that the Galois group of $ f(x)$ over $ \mathbb{Q}$ is the dihedral of order $ 8.$


5. Let $ \mathbb{C}(X)$ denote the rational function field in the indeterminate $ X$ over $ \mathbb{C}.$ Let $ a \in \mathbb{C}$ and $ \sigma_a : \mathbb{C}(X) \rightarrow \mathbb{C}(X) $ be the automorphism that substitutes $ X$ by $ X+a.$ Put $ G=\{\sigma_a: a \in \mathbb{C}\}.$ Show that $ \mathbb{C}(X)^G=\mathbb{C}.$


6. Suppose that the Galois group of a field extension $ K/F$ is the Klein $ 4$-group $ V_4.$ Show that $ K/F$ is biquadratic.


7. Let $ E=\mathbb{Q}(r)$ where $ r$ is a root of $ f(x)=x^3+x^2-2x-1$ in $ \mathbb{C}.$ Show that $ f(r^2-2)=0.$ Determine $ G(E/\mathbb{Q}).$


8. Let $ x,y$ be variables. Let $ a,b,c,d \in \mathbb{Z}$ and $ n=\vert ad-bc\vert.$ Show that $ L=\mathbb{C}(x,y)$ is a Galois extension of $ K=\mathbb{C}(x^ay^b,x^cy^d)$ of degree $ n.$ Find $ G(L/K).$