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 $k[x_{1},x_{2},\ldots ,x_{n}]$ are in bijection with the points in $k^{n}$, where the maximal ideal corresponding to a point $(a_{1},a_{2},\ldots ,a_{n})$ is the ideal $(x_{1}-a_{1},x_{2}-a_{2},\ldots ,x_{n}-a_{n})$
• The max-spectrum of $k[x_{1},x_{2},\ldots ,x_{n}]$, 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 $k[x_{1},x_{2},\ldots ,x_{n}]$ 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 $k^{n}$