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Math Help - Algebraically closed field

  1. #1
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    Algebraically closed field

    Prove that all algebraically closed field is finite.

    thanks!!
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  2. #2
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    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.
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  3. #3
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    Quote Originally Posted by whipflip15 View Post
    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.
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  4. #4
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    Yeah i thought so. There are many ways to show it however the method above is probably the easiest.
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  5. #5
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    It has already been adressed here.
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