# Algebraically closed field

• Nov 7th 2008, 06:41 PM
roporte
Algebraically closed field
Prove that all algebraically closed field is finite.

thanks!!
• Nov 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 \$\displaystyle F=\{a_{1},a_{2},...,a_{n}\}\$ then the polynomial \$\displaystyle (x-a_{1})(x-a_{2})...(x-a_{n})+1\$ does not have a root in F.
• Nov 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 \$\displaystyle F=\{a_{1},a_{2},...,a_{n}\}\$ then the polynomial \$\displaystyle (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.
• Nov 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.
• Nov 8th 2008, 03:18 PM
ThePerfectHacker