Let $p \not = 2$ be an odd prime number. I am trying to understand how to characterize isomorphism classes of quadratic ramified extensions of $\mathbb Q_p$.
Any such extension $E$ is isomorphic to a field of the form $\mathbb Q_p(\alpha)$ where $\alpha$ is the root of a quadratic Eisenstein polynomial $A(X) \in \mathbb Q_p[X]$. Such polynomials (which we can assume to be monic) are given by $$A(X) = X^2 + paX + pu,$$ where $a\in \mathbb Z_p$ and $u \in \mathbb Z_p^{\times}$.
I have read in notes online that by studying the structure of the quotient group $\mathbb Q_p^{\times} / (\mathbb Q_p^{\times})^2$, one can classify all quadratic extensions of $\mathbb Q_p$. It turns out that there are only two isomorphism classes of ramified quadratic extensions of $\mathbb Q_p$. One isomorphism class is given by $E_1 := \mathbb Q_p[\sqrt{p}]$, corresponding to the choice of Eisenstein polynomial $X^2-p$.
Question 1: How exactly does the quotient group $\mathbb Q_p^{\times} / (\mathbb Q_p^{\times})^2$ relate to the problem of constructing quadratic extensions? The notes I mentioned above do not make this process explicit.
Question 2: What would a representative $E_2$ of the other isomorphism class look like?
Question 3: Given an abstract Eisenstein polynomial $A(X)$ as above and $E$ the associated quadratic ramified extension of $\mathbb Q_p$, how can I determine whether $E$ is isomorphic to $E_1$ or $E_2$?
Bonus question: Do we have a formula for the number of isomorphism classes of ramified extensions of $\mathbb Q_p$ of a given degree $d \geq 2$? And similarly, is there a computational method to distinguish between them, generalizing the quadratic case?