Is there a reason as to why they both have similar etymologies? If not, is there a big book of mathematical etymologies? Please include sources, thanks!
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They have nothing to do with each other except insofar as "field" means "a bunch of stuff." You might be interested in consulting this list of earliest known uses of some of the words of mathematics.
In general you shouldn't take mathematical etymology too seriously. It's common for people to use the same name for unrelated things; see, for example, this old MO question.
Qiaochu Yuan
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