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How does one define the direct product of two fields, and is the direct product of two fields itself a field?

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  • $\begingroup$ You can't. There will be zero divisors and the result will not be a field. $\endgroup$ – Kenny Lau Sep 21 '17 at 17:29
  • $\begingroup$ And a field has nothing to do with a vector-field. $\endgroup$ – Kenny Lau Sep 21 '17 at 17:29
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https://en.wikipedia.org/wiki/Direct_product

A direct product of two fields is not a field but a ring.

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