Regarding additive inverse and zero element of a vector space

Question

If zero element doesn't exist in a vector space,then does that mean that additive inverse can also not exist? For example, take a set V of all real polynomials with degree 4 or 6 with the usual scalar multiplication and vector addition, then is V a vector space? If not, then what properties does it not satisfy?

Answer 1

A real vector space $V,+,\cdot$ is a commutative group $V,+$ and there is a scalar multiplication $\cdot:\mathbb{R}\times V\rightarrow V $such that $1⋅v=v$ and it satisfies a left and right distributive identity. The fact that it is a commutative group implies that it must have a neutral element. Thus the space of polynomials of degree 4 and 6 cannot be a vector space as the neutral element is the zero polynomial (whose degree is -1 by convention). Vector spaces in general, do not carry a notion of inverses other than inverses w.r.t. the group structure.

As indicated in the comments, one cannot speak about inverses without the notion of a neutral element first.