ABSTRACT
This section presents some more sophisticated mathematical ideas that allow us to prove more elegantly the impossibility of the constructibility problems of the previous section. This more powerful approach to field theory is interesting in its own right and also plays a vital role in Section IX. First, we define a vector space V over a field F . This is a set of
objects (called vectors) equipped with a binary operation called addition, making the set an additive group. In addition a vector space has a scalar multiplication whereby vectors are multipied by elements (scalars) from the field F . The complete set of axioms follows, where v,w,u ∈ V and r, s ∈ F :
1. v+w = w + v,
2. v+ (w + u) = (v +w) + u,
3. there exists a zero vector 0, with the property that v + 0 = v, and
4. every vector v has an additive inverse −v, with the property that v + (−v) = 0.
