Exercise 111.16.1. A silly argument using the complex numbers! Let ${\mathbf C}$ be the complex number field. Let $V$ be a vector space over ${\mathbf C}$. The spectrum of a linear operator $T : V \to V$ is the set of complex numbers $\lambda \in {\mathbf C}$ such that the operator $T - \lambda \text{id}_ V$ is not invertible.

1. Show that $\mathbf{C}(X)$ has uncountable dimension over ${\mathbf C}$.

2. Show that any linear operator on $V$ has a nonempty spectrum if the dimension of $V$ is finite or countable.

3. Show that if a finitely generated ${\mathbf C}$-algebra $R$ is a field, then the map ${\mathbf C}\to R$ is an isomorphism.

4. Show that any maximal ideal ${\mathfrak m}$ of ${\mathbf C}[x_1, \ldots , x_ n]$ is of the form $(x_1-\alpha _1, \ldots , x_ n-\alpha _ n)$ for some $\alpha _ i \in {\mathbf C}$.

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