The proof proceeds as follows: :$(1): \quad$ showing that the theorem can be captured using a first-order sentences in the language of rings :$(2): \quad$ showing that the theorem is true for at least one field of every characteristic $p > 0$ :$(3): \quad$ applying the Lefschetz Principle (First-Order). === First-Order Sentences in Language of Rings === Since $n$ is fixed, we can quantify over polynomials. There are only finitely many coefficients to quantify over. Therefore each variable occurs with degree at most $d$, say. Thus, for each $n$ and $d$, we can build an $\mathcal L_r$ sentence which holds in a field $F$ {{iff}} every injective polynomial map $F^n \to F^n$ where each variable occurs with at most degree $d$ is surjective. First, we write a formula: : $\phi_{\left({i_1, \dots, i_n}\right)}$ which says that the $n$-variable polynomial map with coefficients $a_{\left({i_1, \dots, i_n}\right)}$ where $(\left({i_1, \dots, i_n}\right) \le \left({d, \dots, d}\right)$ is injective. Note that the polynomial map takes on $n$-tuple images, so is of the form: : $f = \left({f_1, \dots, f_n}\right)$ where each $f_k$ is a polynomial in $n$ variables. The variables $a_{k, \left({i_1, \dots, i_n}\right)}$ in the sentence below are intended to be interpreted as the coefficients of $f_k$: :$\displaystyle \forall x_1 \cdots \forall x_n \forall y_1 \cdots \forall y_n \left({\left({\bigwedge_{k \mathop \le n} \ \sum_{\left({i_1, \ldots, i_n}\right)} a_{k, \left({i_1, \ldots, i_n}\right)} {x_1}^{i_1} \cdots  {x_n}^{i_n} = \sum_{ \left({i_1, \ldots, i_n}\right)} a_{k, \left({i_1, \ldots, i_n}\right)} {y_1}^{i_1} \cdots {y_n}^{i_n} }\right) \to \bigwedge_{i \mathop = 1, \ldots, n} x_i = y_i}\right)$ We also write a formula $\psi_{(i_1,\dots,i_n)}$ which says that such a polynomial is surjective: :$\displaystyle \forall z_1 \cdots \forall z_n \exists x_1 \cdots \exists x_n \left(\bigwedge_{k \mathop \le n} \ \sum_{(i_1, \ldots, i_n)} a_{k, \left({i_1, \ldots, i_n}\right)} {x_1}^{i_1} \cdots {x_n}^{i_n} = z_k \right)$ Finally, we combine these into the required implication, quantifying over all coefficients: :$\displaystyle \underset{k, \left({i_1, \ldots, i_n}\right)} {\huge \forall} a_{k, \left({i_1, \ldots, i_n}\right)} \left({ \phi_{\left({i_1, \ldots,i_n}\right)} \to \psi_{\left({i_1, \ldots, i_n}\right)} }\right)$ Note that we have one of these sentences for every maximum degree $d$ of the variables in a polynomial map. Thus it has been demonstrated that the theorem can be captured using a first-order sentences in the language of rings. {{explain|Is that "a first-order sentence" or "first-order sentences"? It's one or the other. Also in the exposition in the first line.}} {{qed|lemma}} === Field of Characteristic $p > 0$ === Since injections on finite sets are necessarily surjective, every injective polynomial map $k^n \to k^n$ is surjective when $k$ is a finite field. We extend this to the algebraic closure of $k$. This will demonstrate that the sentence above is satisfied by at least one model of the theory of algebraically closed fields of characteristic $p$ for every $p > 0$. Let $\mathbb F^{\operatorname{alg} }_p$ be the algebraic closure of the finite field with $p$ elements. Suppose there is an injective polynomial map: :$f: \left({\mathbb F^{\operatorname{alg} }_p}\right)^n \to \left({\mathbb F^{\operatorname{alg} }_p}\right)^n$ which is not surjective. Let $A$ be the set of coefficients appearing in $f$. Let $\left({z_1, \ldots, z_n}\right) \in \left({\mathbb F^{\operatorname{alg} }_p}\right)^n$ be an element not in the range of $f$. Consider the subfield $k$ of $\mathbb F^{\operatorname{alg}}_p$ generated by the elements of $A$ and the elements $z_1, \ldots, z_n$. We have that: :$\displaystyle \mathbb F^{\operatorname{alg} }_p = \bigcup_{n \mathop = 1, 2, \ldots} \mathbb F_{p^n}$ Therefore any finitely generated subfield is contained in some finite sub-union: :$\displaystyle \bigcup_{n \mathop = 1, 2, \ldots, N} \mathbb F_{p^n}$ Hence $k$ is finite. Therefore, $f {\restriction_{k^n} }$ is an injective polynomial map on a finite field which is not surjective. This is a contradiction. So it must be the case that every injective polynomial map on $\left({\mathbb F^{\operatorname{alg} }_p}\right)^n$ is surjective. That is: :$\left({\mathbb F^{\operatorname{alg} }_p}\right)^n$ satisfies the sentences above for each characteristic $p > 0$. Thus it has been demonstrated that the theorem is true for at least one field of every characteristic $p > 0$. {{qed|lemma}} === Application of Lefschetz Principle === It has been shown that the sentences above are true in some algebraically closed field of characteristic $p$ for all $p > 0$. By the Lefschetz Principle, it follows that they are true in $\C$. {{qed}} {{Namedfor|James Burton Ax|name2=Alexander Grothendieck|cat=Ax|cat2=Grothendieck}} Category:Model Theory Category:Complex Analysis 7e9ntfvhtkr9fox900mtt6h25z9jch0