About $K$-embeddings and the pertinence of an element of a finite and separable field extension Let $L|K$ be a finite and separable field extension of degree $n$, and $\sigma_i:L\to\bar{K},\;i=1,\ldots,n$ the (distinct) $n$ $K$-embeddings of $L$ into some algebraic closure $\bar{K}$ of $K$. By definition, if $\alpha\in K$, then $\sigma_i(\alpha)=\alpha,\;\forall\; i$. Now, suppose that $\sigma_i(\lambda)=\lambda,\;\forall\; i$ for some $\lambda\in L$. Then, is it true that $\lambda\in K$?

The subextension of the algebraic closure generated by $\lambda$ and its images under all embeddings is a finite, normal and separable extension of K, so this follows from the usual theory of Galois extensions.