For example, defining an addition and multiplication on the unit circle in the complex plane such that it forms a field.

This would be a sort of continuous analog of the finite fields. Another way I thought you might construct this is by defining modular equivalence classes on the real numbers by modding the whole part of the number. Does such a "continuous finite field" exist? If not, why can't one exist?

  • $\begingroup$ If you just want the continuum cardinality, as sdcvvc has said, $\mathbb{F}_p(\{x_i\})$ does the trick for a set of countable variables $\{x_i\}$. $\endgroup$ – Eoin May 30 '15 at 21:59
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    $\begingroup$ Regarding the structure of continuous field on the unit circle you can see math.uga.edu/~pete/8410Chapter5.pdf . $\endgroup$ – Alex W May 31 '15 at 0:17

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