Abstract We use quantifier elimination for R and Q p to give elementary proofs of some basic results on algebraic groups G over these fields. In particular (i) the commutator subgroup is closed (and in the real case, semialgebraic), and (ii) if G is simple then any homomorphism of G into a compact topological group is continuous ( van de Waerden, Math. Z. 36 (1933) , 780–786; A. Borel and J. Tits, Ann. of Math. (2) 97 (1973) , 499–571).