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What is an affine n-space?

In some books, k is a field and it says this is just the point space k^n. But on wikipedia there is a whole abstract definition of it, and I don't know whether it corresponds to what algebraic geometry books refer to (there is no mention of fields, and in the books there is no mention of vector spaces, so I wouldn't know how to apply it).

Is it an actual algebraic structure or just a set or?