Algebraically closed field

**roporte** · November 7th 2008, 06:41 PM

Prove that all algebraically closed field is finite.

thanks!!

**whipflip15** · November 7th 2008, 07:33 PM

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.

**roporte** · November 7th 2008, 07:47 PM

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.

**whipflip15** · November 7th 2008, 07:56 PM

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

**ThePerfectHacker** · November 8th 2008, 03:18 PM

It has already been adressed here.

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