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.


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.

  • $\begingroup$ 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
  • 1
    $\begingroup$ @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

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 $\Bbb{C}[x_1, x_2, \dots ]$.

  • $\begingroup$ 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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