# Math Help - Algebraically closed field

1. ## Algebraically closed field

Prove that all algebraically closed field is finite.

thanks!!

2. The complex numbers are algebraically closed but they are infinite...

In fact a finite field cannot be algebraically closed. This is easy to see because if F is a finite field and $F=\{a_{1},a_{2},...,a_{n}\}$ then the polynomial $(x-a_{1})(x-a_{2})...(x-a_{n})+1$ does not have a root in F.

3. Originally Posted by whipflip15
The complex numbers are algebraically closed but they are infinite...

In fact a finite field cannot be algebraically closed. This is easy to see because if F is a finite field and $F=\{a_{1},a_{2},...,a_{n}\}$ then the polynomial $(x-a_{1})(x-a_{2})...(x-a_{n})+1$ does not have a root in F.
SORRY! I must say prove that all algebraically closed field is infinite.

4. Yeah i thought so. There are many ways to show it however the method above is probably the easiest.

5. It has already been adressed here.