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4 Use LaTeX symbol for integers.

[Found another of these Putnam problems; it seems that the protocol here is to post separate big-list examples separately rather than add them to one big-answer.]

2002 Problem B-6. Let $p$ be a prime number. Prove that the determinant of the matrix $$\left(\begin{array}{lll}x&y&z\cr x^p &y^p&z^p\cr x^{p^2}&y^{p^2}&z^{p^2}\end{array}\right)$$ is congruent modulo $p$ to a product of polynomials of the form $ax+by+cz$ where $a,b,c$ are integers.

This actually works for the analogous $n\times n$ determinant for each $n$, and also over any finite field $k$ rather than just the prime field ${\bf Z} \mathbb{Z} / p {\bf Z}$. \mathbb{Z}$. The case$n=2$is basically Fermat's little theorem; the general case is Moore's$q$-analogue of the Vandermonde determinant, used by Dickson to find the subring of$k[x_1,\ldots,x_n]$invariant under all$k$-linear transformations of the$x_i$: they're the polynomials in$n$fundamental invariants of degrees$q^n - q^i$($i=0,1,2,\ldots,n-1$), with the invariant of degree$q^n-1$being the$q-1$power of the Moore determinant. Moreover, replacing that power with the Moore determinant itself yields the invariants for$SL_n(k)$instead of$GL_n(k)$. This century-old theorem has found applications ranging from algebraic topology to Diophantine and algebraic geometry. References: E.H.Moore: A two-fold generalization of Fermat's theorem, Bull. AMS 2 #7 (1986), 189-199. L.E.Dickson: A fundamental system of invariants of the general modular linear group with a solution of the form problem. Trans. AMS 12 (1911), 75-98 Post Made Community Wiki by S. Carnahan 3 Fixed grammar in intro [Found another of these Putnam problems; it seems that the protocol here is to post separate big-list examples separately rather than add in them to one big-answerbig-answer.] 2002 Problem B-6. Let$p$be a prime number. Prove that the determinant of the matrix $$\left(\begin{array}{lll}x&y&z\cr x^p &y^p&z^p\cr x^{p^2}&y^{p^2}&z^{p^2}\end{array}\right)$$ is congruent modulo$p$to a product of polynomials of the form$ax+by+cz$where$a,b,c$are integers. This actually works for the analogous$n\times n$determinant for each$n$, and also over any finite field$k$rather than just the prime field${\bf Z} / p {\bf Z}$. The case$n=2$is basically Fermat's little theorem; the general case is Moore's$q$-analogue of the Vandermonde determinant, used by Dickson to find the subring of$k[x_1,\ldots,x_n]$invariant under all$k$-linear transformations of the$x_i$: they're the polynomials in$n$fundamental invariants of degrees$q^n - q^i$($i=0,1,2,\ldots,n-1$), with the invariant of degree$q^n-1$being the$q-1$power of the Moore determinant. Moreover, replacing that power with the Moore determinant itself yields the invariants for$SL_n(k)$instead of$GL_n(k)\$. This century-old theorem has found applications ranging from algebraic topology to Diophantine and algebraic geometry.

References:

E.H.Moore: A two-fold generalization of Fermat's theorem, Bull. AMS 2 #7 (1986), 189-199.

L.E.Dickson: A fundamental system of invariants of the general modular linear group with a solution of the form problem. Trans. AMS 12 (1911), 75-98