Simple algebraic extensions of a field

Question

Let $K$ be a field, $\overline{K}$ the algebraic closure of $K$, and let $a,b\in\overline{K}\setminus K$ such that $a$ and $b$ are not linear combination of each other with coefficients in $K$. If $K(a)=K(b)$, can we conclude that $a$ and $b$ are conjugate over $K$ ($a$ and $b$ have the same minimal polynomial over $K$)?

Answer 1

No. $\sqrt{2}$ and $\sqrt{2}+1$ generate the same extension over $\mathbb{Q}$, but their minimal polynomials are clearly different.