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I am reading a problem and its solution posted online here that says:

Problem 3: Give an example of a Noetherian ring R that contains a subring that is not Noetherian.

And then,

Solution: The polynomial ring $R = \mathbb C[x_1, x_2, . . .]$ in infinitely many variables is not Noetherian, because the chain of ideals $$(x_1) \subset (x_1, x_2) \subset (x_1, x_2, x_3) \subset . . .$$ does not terminate. On the other hand, this ring is contained in the field $\mathbb C(x_1, x_2, . . .)$, and every field is obviously Noetherian.

My question is simple (but perhaps dumb): I know $\mathbb C[x_1, x_2, . . .]$ means polynomial of infinite indeterminate over complex number $\mathbb C$, but what does $\mathbb C(x_1, x_2, . . .)$ mean in field?

Thank you very much for your time and help.

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You may simply view it as quotient field of the polynomial ring, i.e., it is obtained the same way you obtain $\Bbb Q$ from $\Bbb Z$ by "allowing" to divide by anything but the zero element.

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  • $\begingroup$ Thanks. Is there any other notation beside the above, more widely known? Thanks again. $\endgroup$ – Amanda.M Oct 18 '15 at 20:51
  • $\begingroup$ @A.Magnus: Other notation for what? $\endgroup$ – Eric Wofsey Oct 18 '15 at 21:19
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    $\begingroup$ @A.Magnus No. The above is the only widely known one (if widely known entails known by me). Cf. $\Bbb Q[\pi]$ and $\Bbb Q(\pi)$ as the smalles subring and the smallest subfield of $\Bbb R$ containing (both $\Bbb Q$ and) $\pi$. $\endgroup$ – Hagen von Eitzen Oct 18 '15 at 21:28
  • $\begingroup$ Thanks again for your help. $\endgroup$ – Amanda.M Oct 18 '15 at 23:52
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The following is a very standard notation. Let $A \subseteq K$ be an extension of rings, where $K$ is a field (hence $A$ is a domain), and let $x \in K$ be any element. Then $$A[x]$$ denotes the smallest subring of $K$ containing $A \cup \{ x \}$, while $$A(x)$$ denotes the smallest subfield of $K$ containing $A \cup \{ x \}$. In the particular case when $A$ is a field, we have that $A(x)$ is the fraction field of $A[x]$.

Similarly, if you have any subset $S \subseteq K$, you denote $$A[S]$$ as the smallest subring of $K$ containing $A \cup S$, and $$A(S)$$ denotes the smallest subfield of $K$ containing $A \cup S$.

Following this notation, it is quite clear that $\Bbb{C}(x_1, x_2, \dots )$ denotes the field of fractions of the polynomial ring $\Bbb{C}[x_1, x_2, \dots ]$.

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  • $\begingroup$ I think this is confusing. In the context of the OP's question all you needed to say was the last line: $\mathbb{C}(x_1, x_2, \dots)$ is the field of fractions of $\mathbb{C}[x_1, x_2, \dots]$. The rest of your answer doesn't apply to the OP's question, since the $x_i$ are not particular elements of a larger field $K$. $\endgroup$ – Qiaochu Yuan Oct 18 '15 at 22:04
  • $\begingroup$ They are: the larger field is actually the field $\mathbb{C}(x_1, x_2, \dots )$ itself. $\endgroup$ – Crostul Oct 18 '15 at 22:14
  • $\begingroup$ But the OP's entire problem is that they don't know what field this is! $\endgroup$ – Qiaochu Yuan Oct 18 '15 at 22:15

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