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

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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?