Examples

*1) Every Galois extension of prime degree is cyclic, because every finite group G of prime order $p$ is cyclic (for every element $x \neq 1$ of G is of order $p$, hence generates G).

2) Let $F(X) = X^2 + aX + b$ be an irreducible polynomial in $K[X]$. The only case where $F(X)$ is not separable is that where $K$ is of characteristic 2 and $a = 0$. We shall leave this case aside; let E be an extension of K generated by a root $x$ of $F(X)$. We have $[E : K] = 2$ and $F(X) = (X - x)(X + a + x)$, hence E is a Galois extension of K. The Galois group of E over K is of order 2, hence cyclic.

3) Let $\mathbf{F}$ be a field and $\sigma$ an automorphism of finite order. The field E of invariants of $\sigma$ is also the field of invariants of the cyclic group of order $n$ generated by $\sigma$, and hence (V, p. 66, Th. 3) F is a cyclic extension of degree $n$ of E.

We know (I, p. 50) that every subgroup and every quotient group of a cyclic group is cyclic. Therefore (V, p. 68, Cor. 4), if E is a cyclic extension of a field K, of degree $n$, then every subextension F of E is cyclic over K, and E is cyclic over F. For every divisor d of $n$ there exists a unique subfield F of degree d over K contained in E : for in a cyclic group of order $n$ there exists a unique subgroup of index d.