Expand the completed logarithm of each generator in the Dwork basis and compute the coefficients. Each entry of \(M\) splits as a product of a factor depending only on the row \(j\) — the Bernoulli factor — and a factor of the form \(\eta _a^{\, 2j}\) depending on the column through a Teichmüller node \(\eta _a\). Collecting the row factors into a diagonal matrix exhibits the remaining matrix as a Vandermonde matrix in the nodes \(\eta _a^2\).

The nodes are the values \(\eta _a^2\) for \(2 \le a \le (p-1)/2\); they are pairwise distinct and distinct from \(1\), because the Teichmüller character is injective on \((\mathbb {Z}/p\mathbb {Z})^\times \) and squaring is two-to-one with the representatives \(a\) chosen in a half-system. A Vandermonde determinant in distinct nodes is a product of nonzero differences, hence nonzero.

By theorem 3.3, \(\det M = \det D \cdot \det V\), and \(\det V \ne 0\) by theorem 3.4. Hence \(\det M \ne 0\) exactly when \(\det D = \prod _j D_{jj} \ne 0\), i.e. when every diagonal entry is nonzero.

The factor is the reduction modulo \(p\) of a \(p\)-integral rational expression which is a \(p\)-adic unit multiple of \(B_{2j}/2j\); in the range \(2j \le p-3\), both \(2j\) and the denominator of \(B_{2j}\) are prime to \(p\) (lemma 1.6). So the reduction vanishes exactly when \(p\) divides the numerator of \(B_{2j}\).

Theorem 3.5 reduces the determinant condition to the nonvanishing of every diagonal Bernoulli factor, and theorem 3.6 translates each of those conditions into nondivisibility of the corresponding numerator. The finite index set of the matrix is identified with the classical range \(1\le j\), \(2j\le p-3\).

The completed logarithm is a homomorphism on the relevant units: it turns the exponent product into the corresponding integer combination of the logarithms of the generators, and it sends a \(p\)th power \(y^p\) to \(p\log y\). Equating the two expressions exhibits the combination \(\sum _a e_a \log \varepsilon _a\) as \(p\) times an element of the fixed subalgebra. (The sign \((-1)^s\) is killed by the logarithm.)

Take coordinates in the Dwork basis. The \(j\)th coordinate of the combination \(\sum _a e_a \log \varepsilon _a\) is the \(j\)th entry of the matrix-vector product \(Me\) (before reduction). Divisibility by \(p\) of the combination makes every coordinate divisible by \(p\), i.e. every entry of \(Me\) vanishes after reduction modulo \(p\).

Over the field \(\mathbb {F}_p\), a square matrix with nonzero determinant is invertible, so its kernel is trivial.

Apply theorem 3.8 to the \(p\)th-power relation, then theorem 3.9 converts the logarithmic divisibility into \(Me=0\), and theorem 3.10 forces \(e=0\) modulo \(p\).

Theorem 3.11 is exactly the exponent-vanishing hypothesis of the saturation criterion theorem 2.6.