Prove that a polynomial has infinite roots in if is algebraically closed.
thanks!
If is a non-constant polynomial then it factors into where ( not both zero) are linear polynomials. Thus, it suffices to prove that has infinitely many roots. WLOG say then and so for any value of we can find a value of so that . Thus, the number of solutions to the equation is infinite if . Thus, the last remaining step is to show that is infinite. But if then consider the polynomial , this polynomial has no zeros in . Therefore then cannot be algebraically closed - a contradiction. Therefore and that completes the proof.