Is there a finite field whose additive group is not cyclic?
2026-04-28 14:12:00.1777385520
Is there a finite field in which the additive group is not cyclic?
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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.