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non-archimedean field

Source: miklos schweitzer 2005 q5

August 13, 2021
linear algebraarchimedeancompactness

Problem Statement

Let GL(n,K)GL(n, K) be a linear group over the field K with a topology induced by a non-Archimedean absolute value of the field K. Prove that if the matrix M∈GL(n,K)M \in GL (n, K) is contained by some compact subgroup of GL(n,K)GL(n, K), then all eigenvalues of M have absolute value 1.
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