$D:\mathbb{F}[x]\to \mathbb{F}[x]$ is a group homomorphism of $(\mathbb{F}[x],+)$ into itself such that: $$ D(a_0+a_1 x^1+......+a_n x^n)=a_1+2a_2 x^2+....\text{ etc} $$
In the solution book it is said the image of $\mathbb{F}[x]$ by $D$ is $\mathbb{F}[x]$ with the proof that $$ D(1)=0 $$ and $$ D\left(\frac{1}{i+1}a_ix^{i+1}\right)=a_ix^i, $$ but earlier in the text we only defined $n\cdot a$ for $a$ in $\mathbb{R}$ and $n$ in $\mathbb{Z}$ and, given that, what is the meaning of $1/(i+1)\cdot a_i$? Is that it is some other element in $\mathbb{F}$ such that when multiplied by $I+1$ it should give out ai how can we be confident that such an element exists in $\mathbb{F}$?