Fundamental theorem of Galois theory and $\QQ(\zeta_{43})$

1. Let \(L/K\) be finite Galois with group\[ G=\operatorname{Gal}(L/K). \]The assignments\[ E\longmapsto \operatorname{Gal}(L/E), \qquad H\longmapsto L^H \]give inclusion-reversing mutually inverse bijections between intermediate fields\[ K\subseteq E\subseteq L \]and subgroups \(H\le G\). Moreover,\[ [E:K]=[G:H], \]and \(E/K\) is Galois exactly when \(H\normal G\), in which case\[ \operatorname{Gal}(E/K)\cong G/H. \]

2. Since \(43\) is prime,\[ [\QQ(\zeta_{43}):\QQ]=\varphi(43)=42. \]Every automorphism is determined by\[ \zeta_{43}\longmapsto\zeta_{43}^a, \qquad a\in(\ZZ/43\ZZ)^\times. \]Thus\[ \operatorname{Gal}(\QQ(\zeta_{43})/\QQ) \cong (\ZZ/43\ZZ)^\times. \]The multiplicative group of a finite field is cyclic, so this group is cyclic of order \(42\).

3. A cyclic group has exactly one subgroup of order \(d\) for every divisor \(d\) of its order. Since\[ 42=2\cdot3\cdot7, \]it has\[ \tau(42)=(1+1)^3=8 \]subgroups. Therefore there are \(8\) intermediate fields including the two endpoints \(\QQ\) and \(\QQ(\zeta_{43})\).