Is the set of integers a vector space?

My friend says it's not, because he thinks vectors have to have have at least two elements, otherwise they're scalars. I say that (Z, +, *) satisfies the axioms of a vector space and that's all you need.

What does /sci/ think?

Vector spaces exist "over" fields; specifically, for every vector space V there is a field F such that given any vector v in V and any element c of F, c*v is in V. This is not true of the integers.

(Simpler explanation: Your friend is correct for incorrect reasons)

The integers are a module though, which is similar.