Weak nullstellensatz for algebraically closed fields

Statement

Suppose $k$ is an algebraically closed field. Then, the following equivalent statements hold true:

• Any field $K$, which is finitely generated as a $k$-algebra, must be $k$ itself (i.e. isomorphic to $k$ as a $k$-algebra)
• For any maximal ideal of the polynomial ring in finitely many variables over $k$, the quotient field is $k$
• The maximal ideals in the polynomial ring $k[x_1,x_2,\dots ,x_n]$ are in bijection with the points in $k^n$, where the maximal ideal corresponding to a point $(a_1,a_2,\dots ,a_n)$ is the ideal $(x_1-a_1,x_2-a_2,\dots ,x_n-a_n)$
• The max-spectrum of $k[x_1,x_2,\dots ,x_n]$, with the max-spec topology, is homeomorphic to $k^n$ with the Zariski topology, where the bijection is as described above
• Any proper ideal of $k[x_1,x_2,\dots ,x_n]$ has a nonempty vanishing set in $k^n$
• If a system of polynomial equations over $k$ (in $n$ variables) is consistent (i.e. one cannot derive the equation $0=1$ by manipulating them) then the system has a solution in $k^n$