Let $L/K$ be a Galois extensions. It seems to be true to me, after some tinkering, that some $x\in L$ is a primitive element of the extension $L/K$ iff $\sigma(x)\neq x$ for all $\sigma\in \text{Gal}(L/K)$, where $\sigma$ isn't the identity (though I'm not sure about the "iff" part). How could that be proven ?
Characterisation of primitive elements
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galois-theory