Polynomial Ideals are Subspaces

Corrolary

Let $M$ be an Ideal of $\mathbb{F}[x]$, Then $M$ is a subspace of $\mathbb{F}[x]$

Proof

1. Let $c\in E$
2. Let $g_1,g_2\in M$
3. Then, $c\in \mathbb{F}[x]$
4. Thus, $c{g}_1\in M$
5. Additionally, $c{g}_1+g_2\in M$
6. Thus, $M$ is a subspace by the definition of Alternative Characterization of Polynomial Ideal