every finite field is perfect
- From: warren065@xxxxxxx
- Date: 17 Apr 2005 23:11:25 -0700
A field F of prime characteristic p is said to be perfect if the
mapping a -> a^p isa surjection on F.
1) Show that every finite field is perfect
2) Let F be an arbitrary field of characteristic p not equal to 0.
Show that the field F(x) is not perfect.
Proof: 1) Since we have a surjection and a finite field, we have a
bijection. In a finite field of characteristic p greater than 0, the
kernel of the map is trivial as x^p=x*x^(p-1)=0 over a field makes
x=0.
Can I assume the surjection or do I need to prove that? What would be
a proper proof?
Proof: 2) No clue whatsoever how to approach this.
Help
Thanks,
Warren
.
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