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The difference between [domain] and [field] will depend closely on the context in which each word is used. Occasionally the words will be near-synonyms, more often they will be wholly unrelated.
In reference to land: a [domain] is the area over which a particular feudal officer has legislative authority. An Earl has an earldom as his domain, a Baron has a barony, and a King has a country. A field on the other hand is simply a designated plot of agricultural land (often delimited by hedges and / or a wall).
A mathematical function has a domain, but never a field. We speak of electrical fields, but not domains.
To discover which word to use for a given meaning, you need a dictionary: there is no simple rule of thumb.
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