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Why 10-adics are not a field?

  1. Aug 22, 2011 #1
    I've got a question and I really need the answer! Why 10-adics are not a field? And generally, How can you be sure that a given set is a field or not? For example rational numbers are a field, but what about the others and how can you be sure?
  2. jcsd
  3. Aug 23, 2011 #2


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    You cannot decide whether a given set is a field. You can decide whether a given ring is a field. After all, a field is by definition a special kind of ring, namely a commutative ring in which every nonzero element is invertible. In particular, a field has no zero divisors (i.e. a field is a domain). If you can show that a certain ring has zero divisors, then it is not a field.
  4. Aug 23, 2011 #3
    My bad, you're right. But what is the zero divisors for 10-adics? Is there any example?
  5. Aug 24, 2011 #4
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