I would like some feedback on my proofs. I am trying to prove Theorem 1.10 (Weak Nullstellensatz) in Introduction to Schemes (Ottem & Ellingsrud, v2.0, January 24th, 2023 at 18:17, p.12), specifically point (ii):
Let $k$ be an algebraically closed field, and $\mathfrak{a}$ an ideal in the polynomial ring $k[x_1,\dots,x_n]$.
(i) It holds that $Z(\mathfrak{a})$ is non-empty if and only if $\mathfrak{a}$ is not the unit ideal;
(ii) The maximal ideals in $k[x_1,\dots,x_n]$ are precisely those of the form $(x_1-a_1,\dots,x_n-a_n)$ for $(a_1,\dots,a_n)\in\mathbb{A}(k)$.
I will take (i) for granted and try to prove (ii). It's actually easy to see (using the evaluation map) that every ideal of the form $(x_1-a_1,\dots,x_n-a_n)$ is maximal. The trouble is proving that every maximal ideal $\mathfrak{m}$ is of this form. I made two attempts. Both start by picking a point $a=(a_1,\dots,a_n)\in Z(\mathfrak{m})$, which is not empty by (i). Then:
Attempt 1: Since $f(a)=0$ for all $f\in \mathfrak{m}$, we have that $\mathfrak{m}\subseteq\ker(\textrm{ev}_a)$ where $\textrm{ev}_a$ is the evaluation map at $a$. I will not prove it here but I did prove the fact that $\ker(\textrm{ev}_a)$ is a maximal ideal equal to $(x_1-a_1,\dots,x_n-a_n)$. By maximality of $\mathfrak{m}$, we actually have equality in the set inclusion above.
Attempt 2: Suppose $x_i-a_i$ has an inverse mod $\mathfrak{m}$; then there exists a polynomial $p\in k[x_1,\dots,x_n]$ such that $(x_i-a_i)p\equiv1$ (mod $\mathfrak{m}$), i.e. the polynomial $q=(x_i-a_i)p-1$ lies in $\mathfrak{m}$. But $q(a)\neq 0$, which is a problem because $a$ is a common root of all polynomials in $\mathfrak{m}$. Therefore $x_i-a_i$ does not have an inverse mod $\mathfrak{m}$, which means that $x_i-a_i\in\mathfrak{m}$ since, by maximality of $\mathfrak{m}$, the quotient $k[x_1,\dots,x_n]/\mathfrak{m}$ is a field and the only element without an inverse in a field is zero. Thus $(x_1-a_1,\dots,x_n-a_n)\subseteq\mathfrak{m}$ and in fact equality holds by maximality of $\ker(\textrm{ev}_a)=(x_1-a_1,\dots,x_n-a_n)$.
Admittedly these attempts are very simple and straighforward. Hopefully they're not wrong too. I'm posing this question only because the proof in the book left me confused for some reason and I want to make sure I got things right before going forward, especially regarding something so basic.
As I stated, I'm looking for confirmation that I understand what's going on correctly. I'm also looking for other, better ways of proving this fact. Please keep in mind that I am a newcomer to commutative algebra and algebraic geometry in general so details are very welcome. Thanks!