  We realize the Pauli group $P$ as Galois group of polynomials over the
rational numbers. It is shown by construction that it is possible to find
\textit{pure} polynomials of the form $X^8+k^2$ for $k\neq \lambda^2,
2\lambda^2; k,\lambda \in \mathbb{N}$, that provide an infinite family of such
Galois extensions. This form is also proven to be a necessary condition for a
realization of $P$ by an irreducible \textit{pure} polynomial of degree 8. It
automatically provides a realization of the quaternion group $Q_8$ by pure
polynomials over a quadratic extension of $\mathbb{Q}$, whereas it is shown to
be impossible to realize $Q_8$ over $\mathbb{Q}$ by any pure polynomial of
degree 8. We link the results to Witt's theory of embedding criteria for the
quaternion group via ternary quadratic forms over $\mathbb{Q}$. This provides
for a connection to general realizability criteria for embeddings of some
finite groups into the Pauli group, and the interesting subtleties therein, via
the equivalence of certain quaternion algebras. We thus show how to extend
Witt's result of 1936 to an embedding of $Q_8$ into the Pauli group.
