Problem 1516

To prove that the determinant of the matrix
$$\begin{bmatrix} x & y & z \\ x^p & y^p & z^p \\ x^{p^2} & y^{p^2} & z^{p^2} \end{bmatrix}$$
is congruent modulo $p$ to a product of polynomials of the form $ax + by + cz$, where $a$, $b$, and $c$ are integers, we will use properties derived from Fermat's Little Theorem.

1. Fermat's Little Theorem:
Fermat's Little Theorem states that for any integer $a$ and a prime $p$,
$$a^p \equiv a \pmod{p}.$$
This implies that for any integer $a$ not divisible by $p$,
$$a^{p-1} \equiv 1 \pmod{p}.$$

2. Matrix Setup:
Consider the matrix
$$A = \begin{bmatrix} x & y & z \\ x^p & y^p & z^p \\ x^{p^2} & y^{p^2} & z^{p^2} \end{bmatrix}.$$
We need to show that the determinant of $A$ can be factored into linear factors modulo $p$.

3. Row Operations:
Perform the following row operations on $A$:
- Subtract $x^{p-1}$ times the first row from the second row.
- Subtract $x^{p^2-p}$ times the second row from the third row.

After these operations, the matrix becomes:
$$\begin{bmatrix} x & y & z \\ 0 & y(y^{p-1} - x^{p-1}) & z(z^{p-1} - x^{p-1}) \\ 0 & y^p(y^{p^2-p} - x^{p^2-p}) & z^p(z^{p^2-p} - x^{p^2-p}) \end{bmatrix}.$$

4. Simplification Using Fermat's Little Theorem:
Let $u = y(y^{p-1} - x^{p-1})$ and $v = z(z^{p-1} - x^{p-1})$. Using Fermat's Little Theorem, we know:
$$u^p \equiv y^p(y^{p-1} - x^{p-1}) \pmod{p},$$
$$v^p \equiv z^p(z^{p-1} - x^{p-1}) \pmod{p}.$$

5. Determinant Calculation:
The matrix now looks like:
$$\begin{bmatrix} x & y & z \\ 0 & u & v \\ 0 & u^p & v^p \end{bmatrix}.$$
The determinant of this matrix is:
$$\det(A) = x \cdot (uv^p - vu^p).$$

6. Factorization:
We need to factor $uv^p - vu^p$. Notice that:
$$uv^p - vu^p = uv(v^{p-1} - u^{p-1}).$$
By Fermat's Little Theorem, $v^{p-1} \equiv v \pmod{p}$ and $u^{p-1} \equiv u \pmod{p}$. Thus:
$$v^{p-1} - u^{p-1} \equiv v - u \pmod{p}.$$

7. Final Factorization:
Therefore, the determinant can be written as:
$$\det(A) = x \cdot uv \cdot (v - u).$$
Each of these terms $x$, $u$, $v$, and $v - u$ can be expressed as linear polynomials in $x$, $y$, and $z$.

8. Conclusion:
The determinant of the matrix is congruent modulo $p$ to a product of polynomials of the form $ax + by + cz$, where $a$, $b$, and $c$ are integers.

$$\blacksquare$$