Let \(L/K\) be a finite Galois extension with \[ \operatorname{Gal}(L/K)\cong C_{42}. \] List the proper nontrivial intermediate fields and their degrees over \(K\).
Solution
By the Galois correspondence, intermediate fields correspond bijectively and inclusion-reversingly to subgroups of \(C_{42}\).
A cyclic group has exactly one subgroup of order \(d\) for every divisor \(d\mid42\). The proper nontrivial subgroup orders are \[ 2,3,6,7,14,21. \] If a subgroup has order \(d\), its fixed field \(E\) satisfies \[ [E:K]=[C_{42}:H]=\frac{42}{d}. \] Therefore there is exactly one intermediate field of each degree \[ 21,14,7,6,3,2. \] Equivalently, for every proper nontrivial divisor \(e\) of \(42\), there is a unique intermediate extension of degree \(e\) over \(K\).