What is Galois theory? State the main theorem. What is the splitting field of \(x^5 - 2\) over \(\QQ\)? What are the intermediate extensions? Which extensions are normal, which are not, and why? What are the Galois groups (over Q) of all intermediate extensions?
Solution
Let \[ \alpha=2^{1/5},\qquad \zeta=\zeta_5. \] The roots of \(x^5-2\) are \(\alpha,\zeta\alpha,\ldots,\zeta^4\alpha\), so its splitting field is \[ L=\QQ(\alpha,\zeta). \] By Eisenstein at \(2\), \([\QQ(\alpha):\QQ]=5\). The intersection \(\QQ(\alpha)\cap\QQ(\zeta)\) has degree over \(\QQ\) dividing both \([\QQ(\alpha):\QQ]=5\) and \([\QQ(\zeta):\QQ]=4\). Hence the intersection is \(\QQ\), and therefore \[ [L:\QQ]=20. \]
Define \[ \sigma(\alpha)=\zeta\alpha,\quad \sigma(\zeta)=\zeta, \qquad \tau(\alpha)=\alpha,\quad \tau(\zeta)=\zeta^2. \] Then \[ \sigma^5=\tau^4=1,\qquad \tau\sigma\tau^{-1}=\sigma^2, \] so \[ G:=\Gal(L/\QQ)\cong C_5\rtimes C_4. \]
The fundamental theorem of Galois theory gives an inclusion-reversing bijection \[ \{\QQ\subseteq E\subseteq L\} \longleftrightarrow \{H\le G\}, \qquad E=L^H. \] Moreover \(E/\QQ\) is Galois exactly when \(H\normal G\), in which case \[ \Gal(E/\QQ)\cong G/H. \]
The subgroups of \(G\) are as follows.
1. \(G\) itself fixes \(\QQ\).
2. There is a unique subgroup of order \(10\),\[ H_{10}=\langle\sigma,\tau^2\rangle\cong D_{10}. \]It is normal of index \(2\), and its fixed field is the unique quadratic subfield of \(\QQ(\zeta)\):\[ L^{H_{10}}=\QQ(\sqrt5). \]Thus \(\Gal(\QQ(\sqrt5)/\QQ)\cong C_2\).
3. The unique Sylow \(5\)-subgroup\[ H_5=\langle\sigma\rangle\cong C_5 \]is normal. Its fixed field is\[ L^{H_5}=\QQ(\zeta), \]so\[ \Gal(\QQ(\zeta)/\QQ)\cong C_4. \]
4. There are five conjugate subgroups of order \(4\),\[ H_{4,j}=\sigma^j\langle\tau\rangle\sigma^{-j} \qquad(0\le j<5). \]Their fixed fields are the five conjugate degree-\(5\) fields\[ E_j=\QQ(\zeta^j\alpha). \]None is normal over \(\QQ\). Since each \(H_{4,j}\) is self-normalizing,\[ \Aut_{\QQ}(E_j)\cong N_G(H_{4,j})/H_{4,j}=1. \]
5. There are five conjugate subgroups of order \(2\),\[ H_{2,j}=\sigma^j\langle\tau^2\rangle\sigma^{-j}. \]For \(j=0\) the fixed field is\[ F_0=\QQ\bigl(\alpha,\zeta+\zeta^{-1}\bigr), \]and the other four are its conjugates\[ F_j=\QQ\bigl(\zeta^j\alpha,\zeta+\zeta^{-1}\bigr). \]Each has degree \(10\) over \(\QQ\) and is not normal. The normalizer of \(H_{2,j}\) has order \(4\), so\[ \Aut_{\QQ}(F_j)\cong C_2. \]
6. The trivial subgroup fixes \(L\) itself, and\[ \Gal(L/\QQ)\cong C_5\rtimes C_4. \]
Thus the normal intermediate extensions are exactly \[ \QQ,\qquad \QQ(\sqrt5),\qquad \QQ(\zeta_5),\qquad L, \] while the five degree-\(5\) and five degree-\(10\) intermediate fields are nonnormal.