Hendrik Lenstra says:
Below my 51 errata that I didn't see on your list or in William Stein's mail
yet. Most are of a typographical nature, but some have mathematical
substance. I did at the present occasion not verify the correctness of those.
And: I did not do any proofreading of my list either!! I trust you will apply
your own sound judgment.
Good luck!!
And best regards,
Hendrik
Errata for Cassels-Fr"ohlich
copied by Hendrik Lenstra from his own copy
Jan 13, 2010
Page 3, Proposition 1. This Proposition is misstated, and the proof has the
wrong reference: Chapter II, section 10 has no such result, but Chapter II,
section 5 does. The Theorem in the latter section is the correct formulation:
it is not the extension of the valuation, but the completion that one wants to
be unique. More or less coincidentally, the Proposition is correct as stated
(exercise!), but that statement is neither used (by anybody) nor proved (in the
book).
Page 45, line 5: for "=n" read "=n+1".
Page 52, part (3) of the first definition: for "K" read "V".
Page 54, line -5: replace roman "A" by italic "A" (twice).
Page 73, line 6: replace "vica" by "vice".
Page 75, line 1: replace "(19.9)" by "(19.10)".
Page 78, first line of display (A.19): replace "b_{ij}" by "b_{1j}".
Page 78, line -8 (display (A.24)): replace the third subscript "P" by "R".
Page 78, line -6: replace the third subscript "P" by "R".
Page 79, line -8: remove the three commas within the parentheses.
Page 98, the lower "delta" in the diagram should have a "hat" (the upper
one has one, though it is barely visible in my copy).
Page 123, last line before section 2.5: replace roman "C" by italic "C".
Page 129, line 10: replace "]" by "])".
Page 130, line 1: replace "2.7" by "2.8".
Page 130, line 14: replace "2.5" by "1.5".
Page 131, last line before Corollary 1: replace "2.7" by "2.8".
Page 131, line -10: replace the second "K_{nr}" by "K_{nr}^*".
Page 135, line 6 of Lemma 4: replace the second "M)" by "M))".
Page 139, line 14 (the first display): put ")" before the final ".".
Page 140, line 3 of section 2.3: replace "H^2(G,Z)" by "H^1(G,Q/Z)" (with
Q, Z boldface), since the isomorphism delta hasn't been applied yet!
Page 140, line -8: replace "s" by "s_alpha".
Page 140, line -2: replace "Prop. 2" by "Prop. 1". Also, the proof is
confused. One defines s'alpha to be (alpha,L'/K), and the fact that
salpha maps to s'_alpha under the natural map G^{ab} -> (G/H)^{ab}
FOLLOWS from the equality of character values rather than playing a role
in the proof of that equality.
Page 141, first line after the first diagram: replace the last "K" by "K'".
Page 143, line -3: replace "Lubin" by "Lubin-".
Page 147, first line after Definition: put ")" at the end.
Page 150, proof of Proposition 1: (c) is not proved that way.
Page 150, line -10: for "left-and" read "left- and".
Page 151, line 13: replace the last "[a]" by "[b]".
Page 154, line 18: replace "r_pi" by "r_pi(omega)".
Page 154, line 19: in my copy, there is the scrawled complaint "why is K_pi
from sec. 3.6 equal to K_pi from section 2.8?", and a three line additional
argument, which reads as follows: "Adopt the definition of K_pi as in sec.3.6
(or Theorem 3(b)). Then r(pi) is trivial on K_pi (def. of r), and so is
vartheta(pi) by Cor. to Prop. 6. Also r(pi) and vartheta(pi) are F on
K_{nr}. Hence r and vartheta agree on pi, hence on all of K^*. (Hence also
K_pi=Kpi !)
Page 154, line 2 of section 3.8: replace "2.3" by "2.7".
Page 154, line -8: replace "I_K" by "I'_K".
Page 155, line -11: replace roman "G" by italic "G".
Page 156, line 3: replace "3.3" by "3.4".
Page 156, line 10: replace "beta_j" by "beta^j".
Page 157, line 9: replace "intertia" by "inertia".
Page 158, line -4: replace "|" by "/".
Page 168, line 5: replace roman "F" by italic "F".
Page 168, line -16: replace roman "C" by italic "C".
Page 170, line -18: replace "U^S an arbitrarily small" by "U^S contained in
an arbitrarily small" (because U^S is generally not open).
Page 175, line 2 after the diagram: for "N_{M/K}" read "N_{M/K}J_M".
Page 179, line 12: put ")" before the second "=".
Page 183, line 1: there is no "Proposition 2". Probably "Proposition 2.3" is
meant.
Page 183, display (7): replace the second "K" by "K^*".
Page 192, line -11: replace "infiinte" by "infinite".
Page 211, line -12 (counting the footnote as -1): for "does or does not"
read "does not or does". (This is what I scrawled, I did not verify it at the
present occasion!)
Page 211, line -7: for "seq" read "seq.".
Page 236, line 5: for "2.5", read "1.2, Prop. 1".
Page 353, line 4: for "(lambda,b)_v", read "(b,lambda)_v".
Page 360, last line of Exercise 5.1: for "4.3", read "4.4".
Page 366, under "Tchebotarev, N.,", also list "165," and "227,".
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