Multiplicative subgroups of index three in a field

Leep, David B.; Shapiro, Daniel B.

doi:10.1090/S0002-9939-1989-0963572-X

Multiplicative subgroups of index three in a field
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by David B. Leep and Daniel B. Shapiro PDF

Proc. Amer. Math. Soc. 105 (1989), 802-807 Request permission

Abstract:

Theorem. If $G$ be a subgroup of index 3 in the multiplicative group ${F^*}$ of a field $F$, then $G + G = F$, except in the cases $\left | F \right | = 4,7,13,{\text {or}} 16$. The elementary methods used here provide a new proof of the classical case when $F$ is finite.

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