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

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    @anon what is $C_p^r$ ? Is it is. $Z_p × Z_p × ...× Z_p$ (r times) ?2017-03-24

1 Answers 1

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$F_4$, 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.