A normal uniform algebra that fails to be strongly regular at a peak point

We show that there exists a normal uniform algebra, on a compact metrizable space, that fails to be strongly regular at a peak point.This answers a 32-year-old question of Joel Feinstein.Our example is R(K ) for a certain compact planar set K .Furthermore, our example has a totally ordered oneparameter family of closed primary ideals whose hull is a peak point.We establish general results regarding lifting ideals under Cole root extensions.These results are applied to obtain a normal uniform algebra, on a compact metrizable space, with every point a peak point but again having a totally ordered one-parameter family of closed primary ideals.