Assume we have two field extension $K \subset L $ and $ K \subset M$ assume the last one is Galois. Fix $x\in L$ an algebraic element and $ a \in M$, denote $a_1,\dots ,a_n$ the conjugates of a. Is it true that all the conjugates of $a-x$ over $L$ are $a_1-x,\dots,a_n-x$?

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    $\begingroup$ No. Consider the case of $K=\Bbb{Q}$, $a=x=\sqrt2$. May be you have left out an assumption? $\endgroup$ – Jyrki Lahtonen Jun 19 '18 at 19:40

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