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Is there a finite field where the primitve elements form a additive sub group?

I basically stuck at this problem because primitive elements generate all non zero element of the field so the sub group couldn't be additive without it's 0 element.

At least that is what I think but I'm not even sure about that.

Can someone explain why it is not possible or if possible could you give me an example?

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    $\begingroup$ Indeed, $0$ is not primitive and that's it $\endgroup$ – Hagen von Eitzen May 25 '18 at 6:12
  • $\begingroup$ Thank you for confirming it. $\endgroup$ – Sinka József May 25 '18 at 6:22
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    $\begingroup$ Perhaps it's more interesting to ask whether the primitve elements plus $0$ can form a additive subgroup. $\endgroup$ – lhf May 25 '18 at 11:29

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