Equivalence of Definitions of Algebraically Closed Field

Theorem
Let $K$ be a field. The following are equivalent:


 * 1. The only algebraic extension of $K$ is $K$ itself


 * 2. Every irreducible polynomial $f$ over $K$ has degree $1$


 * 3. Every polynomial $f$ over $K$ of strictly positive degree has a root in $K$

Proof
1. $\Rightarrow$ 2.

Let $f$ be an irreducible polynomial over $K$.

By Principal Ideal of Irreducible Element the ideal $\langle f \rangle$ generated by $f$ is maximal.

So by Maximal Ideal iff Quotient Ring is Field, $L = K[X]/\langle f \rangle$ is a field.

Also $K \subseteq L$, so $L = K$ by hypothesis.