Find a Noether normalization 
Let $I \subseteq k[X_{1}, X_{2}, X_{3}, X_{4}]$ be the ideal generated by the maximal minors of the $2 \times 3$ matrix
$$\begin{pmatrix}
X_1 & X_2 & X_3\\
X_2 & X_3 & X_4
\end{pmatrix}.$$
Find a Noether normalization $k[Y_1, Y_2, Y_3, Y_4] \subseteq k[X_1, X_2, X_3, X_4]$ such that $I \cap k[Y_1, Y_2, Y_3, Y_4] = (Y_1, \ldots, Y_r)$ for a suitable $r$.

What I've done: I never encountered what a maximal minor is, so after some searching I suppose it means the determinant(s) of the maximal submatrices, i.e. in this case all $2 \times 2$ submatrices which then are (by deleting a column): $\begin{pmatrix}
X_1 & X_2\\
X_2 & X_3
\end{pmatrix}, \begin{pmatrix}
X_1 & X_3\\
X_2 & X_4
\end{pmatrix}, \begin{pmatrix}
X_2 & X_3\\
X_3 & X_4
\end{pmatrix}.$ Then, taking determinants we get $I = (X_1 X_3 - X_{2}^{2}, X_1 X_4 - X_3 X_2, X_2 X_4 - X_{3}^{2})$. Is this correct?
The next step would be to use the constructive proof of Noether's Normalization Lemma. However, I can't seem to understand the entire procedure of that proof and how to apply it to this problem. Perhaps if someone can illustrate this process, then I will better understand it after seeing it done.
 A: The function $\mathrm{noethnorml}$ receives

*

*A polynomial ring $R = A[x_1,\ldots,x_m]$ over a coefficient field $A$.

*An ideal $I \subseteq R$ given by its generators.

It returns:
A map of polynomial rings
$$\theta: T_2 = A[w_1,\ldots,w_m] \to R$$
such that

*

*$\theta^{-1}(I) = J = (w_p,\ldots,w_m) \subseteq T_2$ is an ideal of $T_2$. It can be zero (then $p > m$).


*$R/I$ is an integral ring extension of $T_2/J$.
$\mathrm{noethnorml}$ does
A. If $I=0$ then return $\theta = \mathrm{id}_R:R \to R$.
Otherwise do
Select $f \in I$. We want to put $f$ in Noether-Position, that is make a coordinate change, such that $f$ in the new coordinates $y_i$ is of the form
$$(*) y_m^N + a_1 y_m^{N-1} + \cdots + a_m$$
where the $a_i$ are polynomials in $y_1,\ldots,y_{m-1}$.
So define $S=A[y_1,\ldots,y_m]$ and maps
$$\phi: S \to R, \quad y_i \mapsto x_i + \alpha_i x_m \text{ for } i < m \text{ and } y_m \to x_m$$
$$\psi: R \to S \quad x_i \mapsto y_i - \alpha_i y_m \text{ for } i < m \text{ and } x_m \to y_m$$
which are obviously inverse to each other.
The $\alpha_1,\ldots,\alpha_{m-1} \in \mathbb{Z}$ are random integers.
Now for generic $\alpha_i$ the polynomial
$$f_1 = \psi(f)$$
is of the above form $(*)$.
B. Now compute $I_1 = \phi^{-1}(I) \subseteq S$.
C. Define a ring $T=A[u_1,\ldots,u_{m-1}]$ and a map
$$\gamma:T \to S, \quad u_i \mapsto y_i$$
C1. Set $I_2 = \gamma^{-1}(I_1) \subseteq T$.
D. This is the recursion:
Call $\mathrm{noethnorml}$ with arguments $(T, I_2)$.
The result is
$$\theta: T_1 = A[v_1,\ldots,v_{m-1}] \to T$$
E. Call $\rho = \phi \circ \gamma \circ \theta:T_1 \to R$
F. Introduce a ring $T_2 = A[w_1,\ldots,w_m]$ and a ring map
$$\theta':T_2 \to R \quad \text{ with } w_i \mapsto \rho(v_i) \text{ for } i < m \text{ and } w_m \mapsto f$$
G. Return $\theta'$ as the result.
