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.
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.