Are all fields vector spaces?
Solution 1:
A vector space is a set with an addition law and a scalar multiplication law, where the scalars are elements of a field. Thus, a vector space over a field may not be itself a field (e.g. continuous functions on an interval); however, a field is always a vector space over itself. Similarly, taking direct sums of a field will give you a vector space over the original field (e.g $\mathbb{R}^n$ is a vector space over the field $\mathbb{R}$).