TITLE: If there are $k_1, k_2 \in K$ such that $K(\alpha + k_1\beta)=K(\alpha + k_2\beta)$ then $K(\alpha,\beta) = K(\alpha + c\beta)$ for some $c \in K$.
QUESTION [6 upvotes]: Here is the problem: "Let $K \subset M$ be a finite field extension, and $\alpha, \beta \in M$. Suppose there are $k_1, k_2 \in K$ are distinct and such that $K(\alpha + k_1\beta)=K(\alpha + k_2\beta)$. Then $K(\alpha,\beta) = K(\alpha + c\beta)$ for some $c \in K$."
Some help?

REPLY [4 votes]: In fact if $K(\alpha+k_1\beta)$ and $K(\alpha+k_2\beta)$ are the same then they are both $K(\alpha,\beta)$.
Denote the given field by $L$. Hint: from $\alpha+k_1\beta,~\alpha+k_2\beta\in L\,$ deduce $\alpha,\beta\in L$.