1. ## Infinite roots

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

thanks!

2. Originally Posted by roporte
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.