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Try this for the first question. Given an $n$ dimensional scheme of finite type or dimension $n$ noetherian local ring, you can construct an $n$ dimensional local field as explained in chapter $6$ of arxiv.org/pdf/1204.0586.pdf. Notion of $n$ dimensional global field exist ($\mathbb{Q}(t)$ is an example as far as I am aware) but I am not an expert enough to comment much more.
