  We compute the thermally averaged $\qq$--annihilation rate into two and three
gluons in the early universe. We show that at very high temperatures $\qq\to
ggg$ represents only a 3\%\ correction to $\qq\to gg$. Comparing the
annihilation rate to the Hubble rate, corrected for particle interactions in
the Standard Model gas, we find that quarks and gluons are not in chemical
equilibrium when $T\gsim 3\times 10^{14}$ GeV.
