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Let's break down and address each problem in the document step by step.

### Problem 1

Let \(\text{Mat}_2(\mathbb{Z})\) be the ring of \(2 \times 2\) integer matrices and \(R\) be a subring:
\[
\left\{
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix} \in \text{Mat}_2(\mathbb{Z}) \mid a \equiv d \equiv 0 \mod 2024 \text{ and } c \equiv 0 \mod 2024
\right\}.
\]
By viewing \( \text{Mat}_2(\mathbb{Z}) \) as a left module over \( R \) via the natural left action on \( V = \mathbb{Q}^2 \), \( V^+ \) denotes the set of \( R \)-lattices in \( V \) such that \( L \) is generated as an \( R \)-module and \( L \subseteq \mathbb{Z}^2 \). Two \( R \)-lattices \( L \) and \( L' \) are isometric if they are isomorphic as left \( R \)-modules. Find the number of isometry classes of \( R \)-lattices (finite or infinite) and justify your answer.

### Problem 2

Consider the polynomial ring \( \mathbb{C}[x, y] \). An ideal \( I \) is called contraction invariant if for any pair of elements \(\mu, \nu \in \mathbb{C} \setminus \{0\}\), the ideal generated by \((\mu x, \nu y)\) is contained in \( I \). Find the number of contraction-invariant ideals \( I \) in \( \mathbb{C}[x, y] \) with \( \dim(\mathbb{C}[x, y] / I) = 6 \), and justify your answer.

### Problem 3

Let \( n \) be a positive integer and \( F_1, \ldots, F_m \) be homogeneous polynomials in \( \mathbb{C}[X_0, \ldots, X_n] \) such that:
\[
V(F_1, \ldots, F_m) := \{ (x_0, \ldots, x_n) \in \mathbb{CP}^n \mid F_1(x_0, \ldots, x_n) = \cdots = F_m(x_0, \ldots, x_n) = 0 \}
\]
is a finite set. Here, \( \mathbb{CP}^n \) denotes the complex projective space. Prove that the number of elements in \( V(F_1, \ldots, F_m) \) is at most \( d^n \).

### Problem 4

Let \( p > 5 \) be a prime number. Prove that the equation:
\[
\prod_{k=1}^{(p-1)/2} \left( X - 2 \cos \left( \frac{2k\pi}{p} \right) Y \right) = p^2
\]
has no integer solutions.

## Do you need detailed solutions for these problems? Feel free to specify any particular problem you want to dive into further.

### Here are 5 related questions you might want to ask next:

1. How do I approach solving matrix-related algebra problems involving rings and modules?
2. What are contraction-invariant ideals in polynomial rings, and how are they determined?
3. How do you count the number of solutions in complex projective spaces given a set of polynomials?
4. What techniques are useful for proving the non-existence of integer solutions for specific polynomial equations?
5. Can you explain the role of symmetry and invariants in higher-dimensional algebraic problems?

**Tip:** When solving problems in algebra and number theory, breaking down complex statements into smaller, more manageable parts can often reveal insights and simplify the process.

Explore complex problems in ring theory, module theory, and algebraic geometry. Learn how to classify R-lattices, analyze contraction-invariant ideals, and count solutions in complex projective spaces. Suitable for advanced undergraduate or graduate students.

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