A vector field in a Euclidean space is typically represented by a function $F\colon S\to\mathbb{R}^N$, where $S\subset \mathbb{R}^N$.
An algebraic field is a space which has some nice properties.
Is there a mathematical reason both of phrases use the term field? Is there some way we can view one in terms of the other? It looks like the answer is no, but I am not an algebraist. If not, which phrase originated first?
There is no mathematical reason both contain the term field; it's simply etymology.