I am reading Ernst Kunz's "Introduction to commutative algebra and algebraic geometry" and I'm not quite understanding why the underlined sentence in the screenshot below is true. The Nullstellensatz tells us that if $\mathfrak{m}$ is a maximal ideal in $\mathbb{K}[X_1,...,X_n]$ and $\mathbb{K} \subset \mathbb{L}$ with $\mathbb{L}$ algebraically closed, then there is some $x = (x_1,...,x_n)$ with $x \in \mathscr{B}(\mathfrak{m})$. But how can we, from there say that the $x_i$'s are algebraic over $\mathbb{K}$? Can anyone give a hint?
I appreciate all the help and thank you all in advance :)
