How can this statement, with $n\,{\in}\,\mathbb{N}$ and ${\star}:S{\times}S{\rightarrow}S$ being a binary operation on a field $(S,{\star},{\square})$be written in a more formal way? My issue is with the case $n=0$, which I don't know the solution of, and I suppose it could be easier to prove this case with a formal definition of the statement in title.
Context: Exercise 2.
Let's call this $\star^n a$ the expression where there are $n$ instances of $a$ with a star in between each. For $n=0$, we will say $\star^0 a=0_\star$, where $0_\star$ is the element such that $a \star 0_\star=0_\star \star a=a$ for all $a \in S$. For addition, we would have $0_+=0$ and for multiplication, we would have $0_*=1$.
Then, for induction, we will say: $$\star^n a=(\star^{n-1} a)\star a \text{ when } n \geq 1$$ That way, we are adding another $\star a$ as $n$ increases by $1$. This way, we have a formal definition of $\star^n a$.