Non-vanishing of alternants (Q2575021)

Let \(p\) be a prime and \(K\) a field of characteristic \(0\). Let \(x_1\), \(x_2,\dots,x_n\) be non-zero elements of \(K\) such that \(x_i/x_j\) is not a root of unity for \(i\neq j\). Theorem 1 shows that there exist integers \(0<e_1<e_2<\cdots<e_n\) such that \(\det([x_j^{p^{e_i}}])\neq0\). The author deduces from this that each of the sets \[ \Big\{x_1^{p^e},\dots,x_n^{p^e}\mid e\geq0\Big\}\text{ and } \Big\{x_1^{p^e-1},\dots,x_n^{p^e-1}\mid e\geq0\Big\} \] spans \(K^n\) over \(K\). The remainder of the paper consists of applications of Theorem 1 to Witt vectors and to mod \(p\) representations of finite groups. Let \(W(A)\) denote the ring of Witt vectors with coefficients in an integral domain \(A\) of characteristic \(p\). Let \(r:A\mapsto W(A)\) be the canonical multiplicative injection mapping \(a\) to \((a,0,0,0,\dots)\). The author's second theorem is as follows: Let \(v_g\in A\setminus\{0\}\) for each \(g\) in a finite index set \(R\). Suppose \(v_g/v_h\) is not a root of unity in the fraction field of \(A\) for any \(g\neq h\) in \(R\). Then \(\{r(v_g)\mid g\in R\}\) is linearly independent over \(K\).