What does the symbol $\mathbb{R}(\zeta)$ denote?

Question

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?

Answer 1

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