Let denote $P_x$ the minimal polynomial of $x$ over a field $K$. We say that $x$ and $y$ are conjugate if $P_x(y)=0$.
Is "to be conjugate" is an equivalence relation ?
The question behind this question is can $x$ and $y$ be conjugate, $y$ and $z$ be conjugate and $x$ and $z$ not conjugate ? In other term, if $P_x(y)=0$ and $P_y(z)=0$ does $P_x(z)=0$ or not necessarily ?
I don't think it's true, but I can't find any example. Any idea ?
If the minimal polynomial of $x$ is the same as the minimal polynomial for $y$ then $z$ is a root of this polynomial and transitivity holds. But a minimal polynomial is irreducible (by minimality - easily seen because we are over a field so there are no zero divisors) so this is straightforward.
See comments: this depends actually on an assumption about the context. I was taking $x$ to lie in an algebraic extension of the ground field. But the wording of the question admits other possibilities.