How to apply Eisenstein's criterion in complex polynomial

Question

I have two questions

First,

If $f(x) = x + i$, how can I apply Eisenstein's criterion to prove that $f(x)$ is irreducible over $\mathbb{C}[x]$?

Second,

For $f(x, y) = x^2 + y^3$, $f(x,y)$ is irreducible over ($\mathbb{C}[y])[x]$, in which $f(x,y) = x^2 + (-jy)(jy^2)$. Is it correct?

Answer 1

An univariate polynomial over an algebraically closed field is irreducible if and only if its degree is one. So $x+i$ is certainly irreducible. For $x^2+y^3$, this polynomial is irreducible in $(\mathbb C(y))[x]$, since it is a quadric without a root over a field.