Why Are 10-adics Not a Field?

[aalireza]

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?

 

[Landau]

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.

 

[aalireza]

@Landau:
My bad, you're right. But what is the zero divisors for 10-adics? Is there any example?

 

[Citan Uzuki]

See this: http://www.numericana.com/answer/p-adic.htm#composite-radix.