REMARKS AND ERRATA Maximally complete fields

namely the condition that every polynomial of the form a0x pn + a1x pn−1 + · · ·+ an−1xp + anx + an+1 with each ai in the residue field k have a root in k, was shown by Whaples to be equivalent to the condition that k have no extensions of degree divisible by p. See the “Afterthought ” to “Maximal Fields with Valuations ” in [Kap95]. Union-closed families • p. 256, Theorem 1, in condition 2: Change F unionmultiG = G to F unionmultiG ⊆ G. A similar change should be made to the beginning of lines −5 and −3 on p. 257, and to the beginning of line 6 on p. 260. (Thanks to Theresa Vaughan.) Computational aspects of curves of genus ≥ 2 • In the printed version, “positive integer g ” should be changed to “g ≥ 2 ” in the statement of the Shafarevich conjecture in Section 11. The statement “For each number field K and set of places S, there are at most finitely many genus-1 curves over K with good reduction outside S ” implies that the Shafarevich-Tate group of every elliptic curve over a number field is finite. The latter is not yet proved. The number of intersection points made by the diagonals of a regular polygon • The published version contains a typo introduced while texing the manuscript: in Theorem 1, the 232 in the formula for I(n) should be 262, as the routines in ngon.m give. (Thanks to Steve Sommars for noticing this.) The classification of preperiodic points of quadratic polynomials over Q: a refined conjecture • The proof of Proposition 1 in Section 4 contains an error: the point (2,√33) is not on C! Instead (−2,√33) is; apparently a sign got dropped halfway through the computation. To complete the 2-descent correctly one must use also the 2-adic information. The end result of the computation is the same as before, so the main results of the paper still hold. (Thanks to Ken Kramer for noticing the error.