What does the symbol $\mathbb{R}(\zeta)$ denote? I know hat $\mathbb{R}[\zeta]$ denotes the ring of polynomials in $\zeta$ with real coefficients.
I came across the symbol  $\mathbb{R}(\zeta)$. Which ring is this?

The symbol denotes the field of rational functions on $\mathbb{R}$
$\mathbb{R}[\zeta]$ is the ring of polynomials in $\zeta$ over $\mathbb{R}$.
$\mathbb{R} \left({\zeta}\right)$ is a field, and it is defined as follows: $\mathbb{R} \left({\zeta}\right) = \left\{{\forall f \in \mathbb{R} \left[{\zeta}\right], g \in \mathbb{R} \left[{\zeta}\right]^*: \dfrac {f \left({\zeta}\right)} {g \left({\zeta}\right)}}\right\}$.
Showing that this set forms a field requires a proof. Check for example https://proofwiki.org/wiki/Field_of_Rational_Functions
