$(F, +, 0)$
I am looking at my linear algebra lecture notes for next year and immediately came upon this notation I've never seen before, in the context of defining a field. Clearly $F$ is the set, $+$ is the binary operation, and $0$ is the identity, but I'm not exactly sure why we have to specify an identity. Surely if $F$ has an additive identity, we call that $0$, or else $(F, +)$ is not a group. Or is $+$ an arbitrary binary operation but we require the identity to be the Platonic $0$?
This notation is to clarify which symbols we're using. For instance, if we have multiple groups floating around, then it can be nice to explicitly say that $(G, \cdot, e)$ is one group, and $(H, \times, 1)$ is another group. Many logicians and universal algebraists would even specify futher and explicitly include the inverse operation ${}^{-1}$ as part of the structure. So we would have a $4$-tuple $(G,\cdot, {}^{-1}, e)$. Since your author isn't using this, though, I'll also restrict myself to the $3$ tuple in this post.
Even when there's only one group, it can still be good practice to indicate explicitly which symbols you're planning to associate to it. Sure it's usually clear from context, but it doesn't take long to make it explicit.
This becomes more important when you start building structures that can be considered a group in more than one way. For instance, I wouldn't be surprised if the author was trying to be especially clear here because of the two (interacting) group structures $(F,+,0)$ and $(F \setminus \{ 0 \}, \times, 1)$.
I guess it can occasionally confuse a reader (as it did in this case), but you were able to correctly guess what was meant, so I don't think it would have seriously hindered your ability to read the book even if nobody on mse was able to answer the question.
I hope this helps ^_^
