Proposition

Prove that an algebraically closed field $k$ must be infinite.

Solution

Suppose $k = \{ a_1, \cdots, a_n \}$ is finite. Then $f(x) = 1 + \prod_{i=1}^{n} (x - a_i)$ is a polynomial such that $f(a_i) = 1 \ne 0$ for each $i$. Thus $f$ has no roots in $k$. Therefore, $k$ is not algebraically closed.