A Guide to Groups, Rings, and Fields

Fernando Q. Gouvêa

Just published by the MAA in its "Dolciani Guides" series. It exists both as a hardcover book and as an e-book. The second link will allow you to see a "Google preview" of the book.

See below for errata.

If you find more mistakes, please let me know!

Errata

  1. Page 11, line –11: there should be an “if” after “isomorphism”
  2. Page 43, second paragraph, line 2: “ take at G” just be “ take G”.
  3. Pages 52 and 53. I tried to be careful about the conjugation action, in which the acting group G is the same as the set X on which it acts. But I wonder whether some changes might be needed. For example, on page 52, line –5, maybe I should have written “for any x ∈ X = G”. Something similar is definitely need on page 53; I think I would change line 1 to include “and let x be an element of X = G.”
  4. Page 53, line –6: I should have said “let X be a finite set on which G acts”; the case of X empty is actually not a problem, though it's also not very interesting.
  5. Page 54, theorem 4.6.11, item 4: there's probably a space missing between “[Burnside Basis Theorem]” and the statement. Or maybe I just need an italic correction?
  6. Page 59, just below the exact sequence into which the product of two groups fits: The second dotted arrow is certainly a section, but the first is a retraction.
  7. Page 59, line –2: "This one of…" should be "This is one of…"
  8. Page 60, just below the first displayed equation: it is ps, of course, that is the identity; one starts at G2, follows s to G, then returns via p. That's my punishment for writing functions on the left.
  9. In section 4.14.7, it would have been useful to give the standard example of two groups with the same character table, namely D4 and the quaternion group Q.
  10. Page 150, definition 5.7.21: KX should be K[X].
  11. Page 160, first sentence in section 5.9: There is an extra “a”; the correct sentence is “localization is both…”
  12. I love this one. On page 214, following definition 5.6.13, I make a little speech about how one should always say “a gcd” and not “the gcd”. Then read my next sentence!
  13. Page 230, remark between 6.1.23 and 6.1.24, last sentence: As stated this is clearly nonsense. Isomorphism with what? What I think I meant was that if E and F are linearly disjoint and finite dimensional over K, then their tensor product (over K) is isomorphic to their compositum in L, which really boils down to saying that E tensor F is already a field.
  14. Page 240, line –5: The line should start "Because the elements of a transcendence basis…"
  15. Page 280, Reference 60: "Cateogries" should be "Categories".