Let $\mathbb{C} \subseteq K$ be a field and let $a,b,c,d \in K$.
Assume that $k(a+b,c+d)\subseteq k(a,b,c,d)=k(a+b,c+d)(a,c)$ is a finite separable field extension.
By the primitive element theorem, there exists $\lambda \in k$ such that $k(a+b,c+d)(a,c)=k(a+b,c+d)(a+\lambda c)$.
Further assume that:
(1) $\lambda=0$, so $a$ is a primitive element for the extension. (Is it true that then $b$ is also a primitive element for the extension?).
(2) The degree of the field extension is a prime number; see this question. (Is it possible to show that $a$ and $b$ are conjugates= have the same minimal polynomial over $k(a+b,c+d)$?).
Any comments are welcome!