Let $ M $ be a maximal ideal of $ \mathbb{C}[x_{1}, \dots ,x_{n}] $ and let $ \alpha \in \big(\mathbb{C}[x_{1}, \dots ,x_{n}]/M \big) \setminus \mathbb{C} $. I want to show that $ \alpha $ is transcendental over $ \mathbb{C} $.
Since $ \alpha \notin \mathbb{C} $, we can write $ \alpha=f(\overline{x_{1}}, \dots , \overline{x_{n}}) $ for some nonconstant polynomial $ f $. Supposing $ \alpha $ were algebraic over $ \mathbb{C} $, then there would exist a polynomial $ P(T) \in \mathbb{C}[T] $ such that $ P(f(\overline{x_{1}}, \dots , \overline{x_{n}})=\overline{0} $, but I don't know how to derive a contradiction from this though I know it must be simple.
I would appreciate any help. Thank you!