The Pauli group as Galois group of irreducible pure polynomials over Q

Abstract

We realize the Pauli group $P$ as Galois group of polynomials over the rational numbers. It is shown by construction that each pure polynomial in the infinite family of the form $X^8+k^2$ for $k\neq \lambda^2, 2\lambda^2; k,\lambda \in \mathbb{Q}^*$ has Galois group $P$ over $\mathbb{Q}$. This form is also proven to be a necessary condition for a realization of $P$ by any 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 criterion for embedding a biquadratic extension into a normal extension with Galois group $Q_8$ via ternary quadratic forms. This provides for a connection to the known realizability criteria for embeddings of $E_8=C_2^3$-extensions into one with the Pauli group as Galois group, and the interesting subtleties therein, via the equivalence of certain quaternion algebras. We thus show how to exactly extend Witt's result of 1936 to an embedding of $E_8$ into (and realization of $Q_8$ within) the Pauli group.