An Invitation to Arithmetic Geometry

Corrections and Additions

On page 6: Pierre de Fermat (1601-1665). On page 11: In the first paragraph of Example 2.9, a square free polynomial is a polynomial that factors as a product of distinct non-associate irreducible polynomials. In the second paragraph, there is no need to assume k algebraically closed. On page 12, part (iii) of Proposition 2.10: There exists a non-zero finitely generated A-submodule M... On page 33, Exercise I.10. Assume that B is a domain. To see that this condition is necessary, consider the following example. Let A denote the finite field with 2 elements, and let B= A x A. Consider A as a subring of B using the map f(a) = (a,a). The unit in B is (1,1), and the elements (1,0) and (0,1) are integral over A since they are roots of the monic polynomial (1,1) y^2 - (1,1) y. A is a field but B is not. On page 33: Exercise I.16. Assume f(x) of positive degree and not a square in k[x]. On pages 33-34: Exercise I.20. Conclude that $\{1, \xi_p, \dots, \xi_p^{p-2} \}$ is a basis ... $\alpha = \sum_{i=1}^{p-2} a_i \xi_p^i$, with .... $\forall i=0,\dots,p-2$. On page 34: Exercise I.23. Show that $\alpha \in {\overline{\Bbb Q}}^*$... On page 53: Example 5.6. Here is how to factor the singular quartic. On page 68: To apply Zorn's Lemma, we need to check that all totally ordered subsets of $\Sigma$ (i.e. all chains) have an upper bound, and not just the countable ones, as done in the proof of Proposition 8.2. But the proof given in 8.2 in the case of countable chains applies to any chain, thus 8.2 is fully proved. On page 69: Add the following corollary to Proposition 8.2. Corollary: A factorial domain R of dimension 1 is a principal ideal domain. Proof: Lemma I.5.4 shows that all prime ideals of R are principal. Thus we may apply II.8.2. (Compare with I.5.5) On page 69: The analogue of Proposition 8.2, where principal ideal is replaced by finitely generated ideal, is true (see for instance [Mat2], Theorem 3.4). On pages 69-70: The example given in Remark II.8.4 should be replaced by the following one (in the first printing of the book only ). You may also look at the postscript version of the entire remark. Let $R$ denote the subring of the field $k(x,y)$ of rational functions in two variables generated by $k$, $x$, and $y/x^i$, for $i \in \Bbb Z_{\geq 0}$. Let $M_0$ denote the ideal $(x, y/x^i, i \in \Bbb Z_{\geq 0})$. Clearly, $M_0 = (x)$, and it is easy to check that $M_0$ is maximal. Moreover, $\cap^{\infty}_{i=1} M_0^i \supset (y/x^i, i \in \Bbb Z_{\geq 0})$. Let $A := R_{M_0}$, and denote by $M = (x)$ its maximal ideal. On page 82: Exercise II.14. The first part of the exercise is correct only when the hypothesis that $A_i$ is a local ring, for all $i=1,\dots, n$, is assumed. On page 82: Exercise II.20. Let ... $g(x,y) = \sum_{j=0}^m b_j(x) y^j$... On page 83: Exercise II.23. Add the hypothesis that $f$ is irreducible. On page 83: Exercise II.24. Show that if $f(x,y) = \gamma x + \delta y + \sum_{i+j \geq 2} a_{ij} x^i y^j$, ... On page 83, Exercise II.27: The last $V$ should be $V_M$, where $M=(x,y)$ is the ideal corresponding to the point $P$. On page 83, Exercise II.29: ... there exists f(x,y) ... such that C = Z_f(R) or C= Z_f(R) minus (0,0). On page 84, Exercise II.30.b: change the hypothesis $n \neq m$ to $n \leq m$ and $n$ not dividing $m$. On page 91, Proposition III.2.7 states that a local noetherian integrally closed domain R of dimension 1 is a principal ideal domain. Note that this statement is not true in general if R is not assumed to be noetherian. Try to find an example of such an R that is not a principal ideal domain. (Hint: construct R as a well-chosen integral extension of a power serie ring k[[x]], k a field, or look it up .) On page 98, line -2: When A is a principal ideal domain. The words principal ideal need to be added. The point is that in a general domain, an irreducible polynomial f need not generate a prime ideal (f). The hypothesis principal ideal domain can be weakened to normal domain using I.2.17 (communicated by J. Hower, Fall 2005). On page 112, try a different approach to Eisenstein polynomials, as well as a solution to exercise III.39. On page 126, Exercise III.1: The statement is true without the hypothesis that A has dimension 1. This exercise can be applied to show that a factorial domain of dimension 1 is noetherian. On page 127, Exercise III.18, change the name of the field extension $M/L$ to $F/L$. On page 128, Exercise III.20.d: add the hypothesis that $c \leq p$. Remove Exercises III.20.e and III.20.f, and replace them with: Exercise III.20.e: Treat the case $c=b+1$, with $1 \leq b \leq p-2$. (Hint: consider $\pi/y$.) Exercise III.20.f: Treat the case $b \geq c >0$. Show that $(\pi)$ ramifies in $B$ except possibly when $p \mid c$ and $v$ is a $p$-th power modulo $\pi$ Solution. On page 147, proof of Lemma 5.8: the use of the terms `integral basis ' conflicts with the definition given in I.4.11. Replace `integral basis for $B$ over $A$' in 5.8 by `basis of $L$ over $K$ contained in $B$. On page 153, Exercise IV.1: Assume that $f(y) = a_n y^n+ \dots+ a_0$. Show that if $a_n \notin P$, then $\overline{\rm Res}(f,g)}$ equals ${\rm Res}(\overline{f},\overline{g})$ times a power of $\overline{a_n}$. On page 156, Exercise IV.21: Change $S = \cal O \setminus (\pi)$ to $S= \cal O \setminus (0)$. On page 190, Exercise V.12.c: Note that when the characteristic of $k$ is $2$, the ring $k[x][\sqrt{ax^2+b}]$ is not a Dedekind domain. On page 191, Exercise V.23: Add the hypotheses $d>0$ and $n>1$. On page 224, Exercise VI.14: Add the hypothesis that $F$ does not divide $G$. On page 228, line -13. This equality is not always true if the extensions k(a)/k and k(b)/k are not Galois. Only one direction of the claim in 2.4 holds, using the fact that [k(a,b): k(a)] \leq [k(b): k(a) \cap k(b)]. On page 297, line -6: Read ``$Pic^0(X/k)$ is not a finitely generated abelian group'' instead of ``$Pic^0(X/k)$ is not trivial''. On page 302, Exercise VIII.5: Assume that g(X) > 0$. On page 322, line 6: A `minus' sign is missing in front of the second summation sign in this formula for div_c(\alpha). On page 336, Exercise IX.4.a: Change to $F \circ \varphi^{-1} = c (ax^2_0+bx^2_1 x x^2_2)$. (The $c$ is missing in the book.) On page 337, Exercise IX.9.b: `Show that $k'(X)/k$ is a function field' should read `Show that $k'(X)/k'$ is a function field. On page 338, Exercise IX.15: Add the assumption $g>0$. On page 338, Exercise IX.16: Add the hypothesis that $\ell$ is odd. The exercise is sharper if in the subgroup, the exponent $(\ell-1)/2$ is replaced by $\ell-1$. On page 364, last sentence: This conjecture may be due to Lang only. See Lang's letter to the Gazette des Mathematiciens, Octobre 2001, no 90, page 51 and footnote 7.