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Is there a finite field whose additive group is not cyclic?

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For any prime power $q=p^r$ (here $p$ is prime), $\Bbb F_{q}\cong\Bbb F_p[x]/(x^q-x)$, hence $(\Bbb F_q,+)\cong C_p^r$. For $r>1$ this is not cyclic. – anon Aug 25 '12 at 20:17
I’ve found that understanding proceeds from the knowledge of examples. Start with the field ${\mathbb{Z}}/(3)$ with three elements, and formally adjoin the square root of $-1$, i.e. the square root of $2$. Play around in it, splash and frolic, add and multiply, look for an element whose powers exhaust all nonzero elements. You will then know much more than before! – Lubin Aug 25 '12 at 21:21

$F_4$, the Galois field with 4 elements $\{0, 1, \alpha, \beta\}$ has an additive group isomorphic to $V_4$, the Klein four group.

More generally, for a prime $p$ the field with $p^n$ elements for any $n>1$ will not have a cyclic additive group, as any element added to itself $p$ times will be the identity.

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