[Math] Are $P_{>3}$ in the vector space of all polynomials with coefficients

Question

This is the problem statement:

The set of all polynomials of degree greater than 3 together with the zero polynomial in the vector space of $P$ of all polynomials with coefficients in $\mathbb{R}$.

It is not subspace because the given description corresponds to an infinitely generated polynomial space?

Is that correct? If not, how I can solve this?

Answer 1

It is not a subspace because it's not stable by addition: take $F(X)=1-X^4, G(X)=X^4$. Then $F(X)+G(X)=1$ has degree $0$.