Let $\mathbb{k}$ be an algebraically closed uncountable field and $\mathfrak{m}$ a maximal ideal in the polynomial ring $R := \mathbb{k}[x_1, \ldots, x_n]$ in the indeterminates $x_1, \ldots, x_n$. Show that the composite map $\mathbb{k} \longrightarrow R \longrightarrow R/\mathfrak{m}$ is a field isomorphism. You may use without proof the following fact from linear algebra: If a vector space has a countable spanning set, it cannot have a linearly independent uncountable set in it. (Hint: If $t$ is transcendental over $\mathbb{k}$, then consider the set $\left\{\left.\frac{1}{t - \alpha}\right\rvert\, \alpha \in \mathbb{k}\right\}$.)
Let $\mathbb{k}$ be an algebraically closed uncountable field and $\mathfrak{m}$ a maximal ideal in the polynomial ring $R := \mathbb{k}[x_1, \ldots, x_n]$ in the indeterminates $x_1, \ldots, x_n$. Show that the composite map $\mathbb{k} \longrightarrow R \longrightarrow R/\mathfrak{m}$ is a field isomorphism. You may use without proof the following fact from linear algebra: If a vector space has a countable spanning set, it cannot have a linearly independent uncountable set in it. (Hint: If $t$ is transcendental over $\mathbb{k}$, then consider the set $\left\{\left.\frac{1}{t - \alpha}\right\rvert\, \alpha \in \mathbb{k}\right\}$.)