I'm reading some notes that has the following denotation:
- the set of formal power-series with coefficients in $\mathbb{F}_p$ is denoted by $\mathbb{F}_p[[t]]$.
- the fraction field, $\operatorname{Frac}\mathbb{F}_p[[t]]$, is denoted by $\mathbb{F}_p((t))$.
- the fraction field, $\operatorname{Frac}\mathbb{F}_p[t]$, is denoted by $\mathbb{F}_p(t)$.
I'm not sure what the difference in double bracket vs single bracket, and double parenthesis and single parenthesis refers to exactly. I looked for other answers, and found this post, but that hasn't quite elucidated the problem for me. In particular, can somebody help clarify for me what the relationship between $\mathbb{F}_p[[t]]$ and $\mathbb{F}_p[t]$ is?
The ring $\mathbb{F}_p[t]$ consists of polynomials with coefficients in the finite field of order $p$: these are finite expressions of the form $\sum_{i=0}^n a_it^i$, with $a_i \in \mathbb{F}_p$.
In contrast, the ring of formal power series allows for infinite expressions: a typical element looks like $\sum_{i=0}^\infty a_i t^i$. So this ring strictly contains the former one.
The fraction field $\mathbb{F}_p(t)$, which is often called the field of rational functions, consists of all quotients $P(t)/Q(t)$ of polynomials, except when $Q(t) = 0$ is the null polynomial.
Finally, the fraction field $\mathbb{F}_p((t))$ may be called the field of formal Laurent series. A generic element in that field takes the form $\sum_{i \in \mathbb{Z}} a_i t^i$, where all but finitely many terms with negative exponent vanish.