Hi. Are there nice/simple examples of cyclic extensions $L/K$ (that is, Galois extensions with cyclic Galois group) for which $L$ cannot be written as $K(a)$ with $a^n\in K$?


Thanks.



Cyclic extensions
Dear Justin, your question is subtly ambiguous.


**First interpretation:** Is there a cyclic extension $L/K$ of degree $n$ that cannot be written $L=K(a)$ with $a\in L$ and $a^n\in K ?$.


**Answer** Yes. For example, let $p$ be a prime number and $n$ an integer.The extension 
$\mathbb F\_p\subset \mathbb F\_{p^{p^n}}$ is cyclic and not of the required form since $a^{p^n} \in \mathbb F\_p$ implies $a\in \mathbb F\_p .\;$ So the simplest possible example for your question is $\mathbb F\_2 \subset \mathbb F\_4$ ! [Franz and David give excellent examples in characteristic zero]


**Second interpretation:** Is there a cyclic extension $L/K$ of degreee $n$ that cannot be written $L=K(a)$ with $a\in L$ and $a^N\in K$ *for some N that might be different from n ?*


**Partial answer** Some examples in the preceding interpretation disappear! For instance if you take
$\mathbb F\_2 \subset \mathbb F\_4$, you CAN write $\mathbb F\_4= \mathbb F\_2(a)$ with $a^\textbf{3}=1$ : just take for $a$ one of the two elements in $\mathbb F\_4 \setminus \mathbb F\_2$. Actually if $K\subset L$ are finite all examples disappar: all such extensions are cyclic and can be written $L=K(a)$ with $a^N\in K$. Indeed if $a\in L$ is a primitive element (which always exists: our extension is separable), we have $L=K(a)$ and if $q=card(L)$ we can be sure that $a^{q-1}=1 \in K$.


