# Example for Etale Morphism

I'm want to understand the concept of etale morphism of schemes using following definition:

A morphism $$f: X \to Y$$ is etale iff it is * flat *(1), locally of finite type(2) and has the separable field condition(3):

Here (1), (2) and (3) mean:

(1) For every $$x \in X$$ the induced morphism $$f_x ^{\#}:\mathcal{O}_{Y, f(x)} \to \mathcal{O}_X$$ is flat

(2) There exist an open affine neighbourhood $$U_x =Spec(R)$$ of $$x$$ and an o. a. n. $$V_{f(x)}= Spec(S)$$ of $$f(x)$$ with $$f(U_x) \subset V_{f(x)}$$ such that the induced ring map $$S \to R$$ is of finite presentation

(3) Let $$m_x \subset \mathcal{O}_{X,x}$$ the unique maximal ideal of local ring $$\mathcal{O}_{X,x}$$ and respectively $$m_{f(x)} \subset \mathcal{O}_{X,x}$$ the unique maximal ideal of $$\mathcal{O}_{Y,f(x)}$$: Then the induced finite field extension $$\mathcal{O}_{Y,f(x)}/m_{f(x)} \to \mathcal{O}_{X,x}/m_x$$ is separable

I'm trying to acquire the intuition for etaleness by considering following four examples

(a) $$Spec(\mathbb{C}[T, T^{-1}]) \to Spec(\mathbb{C}[T])$$

(b) $$Spec(\mathbb{C}[T] /(T^d - 2))\to Spec(\mathbb{C}[T])$$

(c) $$Spec(\mathbb{C}[T, Y]/(Y^d - T)) \to Spec(\mathbb{C}[T])$$

(d) $$Spec(\mathbb{C}[T, T^{-1},Y]/(Y^d - T)) \to \mathbb{C}[T]$$

My attempts:

(a) Is is flat since it just a localization of $$\mathbb{C}[T]$$ on $$T$$. Or a secound argument: Open embeddings are etale.

But what about (b), (c) and (d)?