I found this statement in a book introducing nonstandard analysis. It also says that “For example, let $E$ be the set of even natural numbers, and let $O$ be the set of odd natural numbers. The product of characteristic functions $\chi _E \cdot \chi _ O$ is identically 0, and should represent $0$. To have a field, therefore, one of the sequences $\chi _E$ or $\chi _O$ must also represent $0$.”
I haven’t learned representation theory (what I think it’s referring to) but I want to kinda understand this motivation in the book, without reading the whole chapters on representation theory (or just some basic algebra?). So can anyone point out a theorem or something that could quickly make me get the overall idea?
To answer the question in your title: any sequence that contains $0$ doesn't have a multiplicative inverse. But in a field, every element except $0$ (which in this case would be the sequence of all $0$'s) must have a multiplicative inverse.
The "For example..." is essentially the case where the sequences are $0,1,0,1,0,\ldots$ and $1,0,1,0,\ldots$. The pointwise product of these is $0,0,0,0,0,\ldots$, which would be the $0$ element of the field. But in a field, the product of two elements can't be $0$ unless one of them is $0$, and that's not the case here.