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 $$k^n$$