Let $k$ be a field of characteristic zero, and let $a,b$ be algebraic over $k$, with minimal polynomials $f_a,f_b$, respectively, of the same prime degree $p \geq 3$. Denote: $f_a=T^p+a_{p-1}T^{p-1}+\cdots+a_1T+a_0$ and $f_b=T^p+b_{p-1}T^{p-1}+\cdots+b_1T+b_0$. Further assume that $a-b$ has also minimal polynomial of degree $p$ over $k$, denote it by $f_{a-b}=T^p+c_{p-1}T^{p-1}+\cdots+c_1T+c_0$. Further assume that $k(a)=k(b)=k(a-b)$.
Is there an explicit connection between the three minimal polynomials?
Thank you very much!
