Motivation: Take an algebraic $\lambda$. In my research, I've stumbled upon the question in which cases the expression $\sum_{\sigma \in S} \sigma(\lambda)$, where $S$ is a subset of the field embeddings of $K=\mathbb{Q}(\lambda)$, can be 'less irrational' then $\lambda$ itself (in the sense that the field generated by it has smaller degree than $K$).

Question: Take a finite Galois extension $L/\mathbb{Q}$ with Galois group $G$, let $H \leq G$ be a subgroup and let $K$ be the field fixed by $H$. Consider a subset $S$ of $G/H$ and consider the $\mathbb{Q}$-linear vector space homomorphism $$ f_S: K \to L, x \mapsto \sum_{ \sigma \in S} \sigma(x). $$ If $SH$ is a subgroup of $G$ strictly bigger than $H$, this map will not be injective because $f_S$ takes values in the subfield of $K$ fixed by $SH$. Intuitively, it seems likely that $SH$ being (the coset of) a subgroup is the only case in which this 'reduction of irrationality' can happen. Is this true, i.e. is $f_S$ injective under the condition that $SH$ is not the coset of a subgroup?