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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?

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    $\begingroup$ It's probably a field extension. $\endgroup$ – Daniel R May 30 '14 at 14:05
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    $\begingroup$ It is given as an example of commutative ring. $\endgroup$ – user144410 May 30 '14 at 14:06
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    $\begingroup$ Usually is the field of fractions of $\mathbb{R}[\zeta]$. $\endgroup$ – Martín-Blas Pérez Pinilla May 30 '14 at 14:10
  • $\begingroup$ Unless the symbol $\;\zeta\;$ is clearly defined, $\;\Bbb R[\zeta]\;$ is just a common notation for ring of polynomials "in that thing" over the reals...and $\;\Bbb R(\zeta)\;$ is that ring's fractions field, as already mentioned...assuming $\;\Bbb R[\zeta]\;$ is an integer domain, of course. $\endgroup$ – DonAntonio May 30 '14 at 14:11
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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

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