Is there an integral domain containing exactly 10 elements?
As always, completely justify your answer.
You may not use the fact that every finite field has prime power order.
Solution
Let x be a nonzero element of the domain E, and consider the multiplication map f : E to E, by
f(y) = xy. Since E is a domain, if xy = 0, then x = 0. Hence, f is injective, and since it is finite it
is also surjective. So, 1 is in f(E); that is, there is some y in E such that xy = 1. So x is a
invertible and D is a field.
Since the only fields of finite have order p^k, we know that an integral domain with 10 elements
does not exist..
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Is there an integral domain containing exactly 10 elementsAs alwa.pdf
1. Is there an integral domain containing exactly 10 elements?
As always, completely justify your answer.
You may not use the fact that every finite field has prime power order.
Solution
Let x be a nonzero element of the domain E, and consider the multiplication map f : E to E, by
f(y) = xy. Since E is a domain, if xy = 0, then x = 0. Hence, f is injective, and since it is finite it
is also surjective. So, 1 is in f(E); that is, there is some y in E such that xy = 1. So x is a
invertible and D is a field.
Since the only fields of finite have order p^k, we know that an integral domain with 10 elements
does not exist.
