= Solution
\b[Yes: $Q^+$ is prime.] In any <first-order model>[model] of its <elementary diagram of a structure>[complete diagram], map each rational to the interpretation of its named constant. The diagram ensures this is an <structure embedding>[embedding], and <quantifier elimination for dense linear orders without endpoints> makes it elementary. Alternatively every finite tuple of $Q^+$ has its <complete type> isolated by the <conjunction> equating its coordinates to their named rational constants. Thus it is a <countable> <atomic model>, and the earlier prime-model equivalence applies. A <prime model> may omit many types; primeness is not saturation.
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