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We realize the symplectic group $Sp(2m,q)$ as a Galois group over thefunction field $\mathbb{F}_q(t)$, where $q$ is a power of the prime $p$. Ourproof is based on the study of certain so-called $q$-palindromic polynomials,whose roots form a vector space over $\mathbb{F}_q$. The main thrust of ourapproach is to find symplectic transvections in the Galois group and to exploitthe fact that the Galois group acts transitively on the nonzero roots in theirreducible case.

Realizing the symplectic group as a Galois group over the function field $\mathbb{F}_q(t)$