Additive Properties of Multiplicative Subgroups of Finite Index in Fields
Pedro Berrizbeitia · Proceedings of the American Mathematical Society · 1991
Gallai’s theorem, an $n$-dimensional generalization of Van der Waerden’s theorem on arithmetic progression, is used to prove the following theorem: Let $F$ be a field and $G \subseteq {F^ * }$ a subgroup of finite index $n$. There is a positive integer $N$, which depends only on $n$, so that if ${\text {Char}}F = 0$ or ${\text {Char}}F \geq N$, then $G - G = F$.