We study properties of the Golomb topology on polynomial rings over fields, in particular trying to determine conditions under which two such spaces are not homeomorphic. We show that if $K$ is an algebraic extension of a finite field and $K'$ is a field of the same characteristic, then the Golomb spaces of $K[X]$ and $K'[X]$ are homeomorphic if and only if $K$ and $K'$ are isomorphic.

中文翻译：

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多项式环的哥伦布拓扑

我们研究了场上多项式环上哥伦布拓扑的性质，特别是试图确定两个这样的空间不是同胚的条件。我们证明如果 $K$ 是有限域的代数扩展并且 $K'$ 是具有相同特征的域，那么 $K[X]$ 和 $K'[X]$ 的哥伦布空间是同胚的，如果并且仅当 $K$ 和 $K'$ 是同构的。