Let $K$ be a number field, $overline{K}$ an algebraic closure of$K$ and $E/K$ an elliptic curvedefined over $K$. In this paper, we prove that if $E/K$ has a$K$-rational point $P$ such that $2P eq O$ and $3P eq O$, thenfor each $sigmain Gal(overline{K}/K)$, the Mordell--Weil group$E(overline{K}^{sigma})$ of $E$ over the fixed subfield of$overline{K}$ under $sigma$ has infinite rank.
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机译：假设$ K $是数字字段，$ overline {K} $是$ K $的代数闭包，而$ E / K $是在$ K $上定义的椭圆曲线。在本文中，我们证明如果$ E / K $具有$ K $理性点$ P $使得$ 2P eq O $和$ 3P eq O $，则对于每个$ sigmain Gal（overline {K} / K ）$，在$ sigma $下的$ overline {K} $固定子字段中，$ E $的Mordell-Weil组$ E（overline {K} ^ {sigma}）$具有无限排名。
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