Prove that all algebraically closed field is finite.

thanks!!

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- Nov 7th 2008, 06:41 PMroporteAlgebraically closed field
Prove that all algebraically closed field is finite.

thanks!! - Nov 7th 2008, 07:33 PMwhipflip15
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 PMroporte
- Nov 7th 2008, 07:56 PMwhipflip15
Yeah i thought so. There are many ways to show it however the method above is probably the easiest.

- Nov 8th 2008, 03:18 PMThePerfectHacker
It has already been adressed here.