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We show that for nonarchimedean local fields $F$, the pairing from thealgebraic part of the Brauer group of a reductive group $G$ characterizes allcontinuous homomorphisms from $G(F)$ into $\mathbb{Q}/\mathbb{Z}$. Thisgeneralizes results of Loughran and Loughran-Tanimoto-Takloo-Bighash.

The algebraic Brauer group of a reductive group over a nonarchimedean local field