Algebraic closed field has infinite many elements

Question

Show that an algebraic closed field must have infinite many elements.

Let's suppose that an algebraic closed field $K$ has finite many elements.

But how could I get a contradiction??

Answer 1

Suppose $K=\{a_1, \ldots ,a_n\}$, for some natural number $n$.

Consider the polynomial $f(x)=(x-a_1)(x-a_2)\ldots(x-a_n)+1$.

Since you assumed that $K$ is algebraically closed, there is a root $a$ of $f(x)$ in $K$, that is $f(a_i)=0$, for some $i\in \{1, \ldots ,n\}$.

Try to get a contradiction.