The matrix Euler–Fermat theorem

We prove many congruences for binomial and multinomial coefficients as well as for the coefficients of the Girard–Newton formula in the theory of symmetric functions. These congruences also imply congruences (modulo powers of primes) for the traces of various powers of matrices with integer elements. We thus have an extension of the matrix Fermat theorem similar to Euler's extension of the numerical little Fermat theorem.