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Math Help - Infinite roots

  1. #1
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    Infinite roots

    Prove that a polynomial f \in K[X,Y] has infinite roots in K^2 if K is algebraically closed.

    thanks!
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  2. #2
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    Quote Originally Posted by roporte View Post
    Prove that a polynomial f \in K[X,Y] has infinite roots in K^2 if K is algebraically closed.

    thanks!
    If f \in K[X,Y] is a non-constant polynomial then it factors into f = \prod_i l_i where l_i = a_iX+b_iY+c_i ( a_i,b_i not both zero) are linear polynomials. Thus, it suffices to prove that l_i has infinitely many roots. WLOG say a_i\not = 0 then X = -a_i^{-1}b_iY - a_i^{-1}c_i and so for any value of Y\in K we can find a value of X\in K so that a_i X + b_i Y + c_i = 0. Thus, the number of solutions to the equation l_i = 0 is infinite if |K| = \infty. Thus, the last remaining step is to show that K is infinite. But if |K| = n then consider the polynomial X^n - X + 1 \in K[X], this polynomial has no zeros in K. Therefore then K cannot be algebraically closed - a contradiction. Therefore |K| = \text{ ME } and that completes the proof.
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