No finite field is algebraically closed. Consider the field of q elements (q some prime power). Then every field element satisfies the equation , and so the polynomial has no roots in the field.
Does there exists a finite field such is algebraically closed?
I tried of course the most obvious example but it is not algebraically closed for has no zero.
This problem seems strongly connected with number theory because it invloves prime numbers because all finite fields have order .
I was thinking about this today and I have a different approach.
(For simplicity sake I am working with only)
1)Every finite field has form .
2)Assume that is closed.
3)Then
4)Thus, .
5)Thus, .
6)Thus, for all (Legendre symbol).
7)Thus, .
8)But, (a theorem)
9)A contradiction.
10)Thus, no finite field of form is algebraically closed.
Now perhaps it can be completed for all field of