Algebraically closed field

Printable View

November 7th 2008, 06:41 PM
roporte

Algebraically closed field

Prove that all algebraically closed field is finite.

thanks!!
November 7th 2008, 07:33 PM
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.
November 7th 2008, 07:47 PM
roporte

Quote:

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.
November 7th 2008, 07:56 PM
whipflip15

Yeah i thought so. There are many ways to show it however the method above is probably the easiest.
November 8th 2008, 03:18 PM
ThePerfectHacker

It has already been adressed here.

All times are GMT -8. The time now is 05:45 AM.