Proposition. equivalent definitions of algebraically closed fields [1601]

    For a field $\Omega$, the following statements are equivalent:

    1. Every non-constant polynomial in $\Omega[x]$ splits completely in $\Omega$.
    2. Every non-constant polynomial in $\Omega[x]$ has a root in $\Omega$.
    3. The irreducible polynomials in $\Omega[x]$ are those of degree $1$.
    4. If $K / \Omega$ is an algebraic extension, then $K = \Omega$.
    5. If $K / \Omega$ is a finite extension, then $K = \Omega$.

    Proof. sketch [1601A]

      1 $\iff$ 2 $\iff$ 3 is obvious.

      3 $\implies$ 4: Take $\alpha \in K$ and consider $m_{\alpha, \Omega}(x)$. It must be $m_{\alpha, \Omega}(x) = x - \alpha \in \Omega[x]$ since it is irreducible and monic, hence $\alpha \in \Omega$. So we conclude that $K = \Omega$.

      4 $\implies$ 5: Obvious because of Proposition 14010.

      5 $\implies$ 1: Take any $f(x) \in \Omega[x]$ and let $K$ be its splitting field. Then by Proposition 1507, $[K : \Omega]$ is at most $n!$ where $n = \deg f(x)$, hence it is finite. So $K = \Omega$. Then we can conclude that every non-constant polynomial in $\Omega[x]$ splits completely in $\Omega$.