Suppose $K, L$ are fields and $\sigma_{1}, \ldots, \sigma_{m}$ are distinct embeddings of $K$ into $L$. Prove that there do not exist elements $\lambda_{1}, \ldots, \lambda_{m}$ of $L$ (not all zero) such that $\lambda_{1} \sigma_{1}(x)+\ldots+$ $\lambda_{m} \sigma_{m}(x)=0$ for all $x \in K$. Deduce that if $K / k$ is a finite extension of fields, and $\sigma_{1}, \ldots, \sigma_{m}$ are distinct $k$-automorphisms of $K$, then $m \leqslant[K: k]$.

Suppose now that $K$ is a Galois extension of $k$ with Galois group cyclic of order $n$, where $n$ is not divisible by the characteristic. If $k$ contains a primitive $n$th root of unity, prove that $K$ is a radical extension of $k$. Explain briefly the relevance of this result to the problem of solubility of cubics by radicals.