Embedding of a cubic field into $\mathbb{R}$ I'm attempting problem IV.3B in Samuel Pierre's Algebraic Theory of Numbers. The problem is as follows:

Let $K$ be a cubic field such that $r_1 = r_2 = 1$. Suppose $K$ is
  imbedded in $\mathbb{R}$.
(a) Show that the positive units of $K$ form a group isomorphic to
  $\mathbb{Z}$, and that every positive unit of $K$ is of norm $1$.
(b) Let $d$ be the absolute discriminant of $K$ and let $u$ be a unit
  greater than $1$. Show that $|d| \le 4u^3 + 24$ (put $u = x^2$ with $x
> \in \mathbb{R}$ and $x > 0$; note that the conjugates of $u$ are of
  the form $x^{-1}e^{ty}$ and $x^{-1}e^{-iy}$ with $y \in \mathbb{R}$.
  Calculate the discriminant $d' = D(1, u, u^2)$ as a function of $x,
> y$, say $|d'|^{1/2} = \phi(x, y)$. Find the minimum of $\phi(x, y)$
  for fixed $x$ and deduce that $|d'| \le 4 u^3 + 24$. Conclude by
  observing that $d$ divides $d'$.)
(c) Show that the polynomial $x^3 + 10x + 1$ is irreducible over
  $\mathbb{Q}$ (cf. Chapter V).

I'm confused by the following things. 
(1) Why can we put $u = x^2$ (or why particularly square)? Why is the conjugate of $u = x^2$ in the form of $x^{-1}e^{iy}$ and $x^{-1}e^{-iy}$?
(2) Is finding the $D(1, u, u^2)$ the same as calculating the determinant of $[\sigma_i(z_i)]$ where $\sigma_i$ ranges over all the embeddings, and $z_i$ ranges over $1, u, u^2$? 
(3) Conceptually what does the discriminant tell us? And why should we find the minimum to deduce the result stated in the problem statement? 
Many thanks to any help! 

(1) every positive real has a positive square root. Also if $u_1$
and $u_2$ are the other conjugates of $u_2$ then $u_2=\overline{u_1}$
and $uu_1u_2=1$. Therefore $|u_1|=x^{-1}$ etc.
