Exercises

1. Let $A=\left( \begin{matrix}4&7&2\ 2&4&6\ 0&0&0 \end{matrix}\right)$.
   1. Find invertible integer matrices $P$ and $Q$ such that $PAQ$ is in Smith normal form.
   2. What is the group structure of the cokernel of the map $\mathbf{Z}^3\to \mathbf{Z}^3$ defined by multiplication by $A$?

2. Let $G$ be the abelian group generated by $x,y,z$ with relatoins $2x+y=0$ and $x-y+3z=0$. Find a product of cyclic groups that is isomorphic to $G$.

3. Prove that each of the following rings have infinitely many prime ideals:
   1. The integers $\mathbf{Z}$. [Hint: Euclid gave a famous proof of this long ago.]
   2. The ring $\mathbf{Q}[x]$ of polynomials over $\mathbf{Q}$.
   3. The ring $\mathbf{Z}[x]$ of polynomials over $\mathbf{Z}$.
   4. The ring $\overline{\mathbf{Z}}$ of all algebraic integers. [Hint: Use Zorn's lemma, which implies that every ideal is contained in a maximal ideal. See, e.g., Prop 1.12 on page 589 of Artin's Algebra.]

4. (This problem was on the graduate qualifying exam on Tuesday.) Let $\overline{\mathbf{Z}}$ denote the subset of all elements of $\overline{\mathbf{Q}}$ that satisfy a monic polynomial with coefficients in the ring $\mathbf{Z}$ of integers. We proved in class that $\overline{\mathbf{Z}}$ is a ring.
   1. Show that the ideals $(2)$ and $(\sqrt{2})$ in $\overline{\mathbf{Z}}$ are distinct.
   2. Prove that $\overline{\mathbf{Z}}$ is not Noetherian.

5. Show that neither $\mathbf{Z}[\sqrt{-6}]$ nor $\mathbf{Z}[\sqrt{5}]$ is a unique factorization domain. [Hint: Consider the factorization into irreducible elements of $6$ in the first case and $4$ in the second. A nonzero element $a$ in a ring $R$ is an irreducible element if it is not a unit and if whenever $a=qr$, then one of $q$ or $r$ is a unit.]

6. Find the ring of integers of each of the following number fields:
   1. $\mathbf{Q}(\sqrt{-3})$,
   2. $\mathbf{Q}(\sqrt{3})$, and
   3. $\mathbf{Q}(\sqrt[3]{2})$.
   Do not use a computer for the first two.

7. Find the discriminants of the rings of integers of the numbers fields in the previous problem. (Do not use a computer.)

8. Let $R$ be a finite integral domain. Prove that $R$ is a field. [Hint: Show that if $x$ is a nonzero element, then $x$ has an inverse by considering powers of $x$.]

9. Suppose $K\subset L \subset M$ is a tower of number fields and let $\sigma:L\hookrightarrow \overline{\mathbf{Q}}$ be a field embedding of $L$ into $\overline{\mathbf{Q}}$ that fixes $K$ elementwise. Show that $\sigma$ extends in exactly $[M:L]$ ways to a field embedding $M\hookrightarrow \overline{\mathbf{Q}}$.

10. 1. Suppose $I$ and $J$ are principal ideals in a ring $R$. Show that the set $\{ab : a \in I, b\in J\}$ is an ideal.
    2. Give an example of ideals $I$ and $J$ in the polynomial ring $\mathbf{Q}[x,y]$ in two variables such that $\{ab : a \in I, b\in J\}$ is not an ideal. Your example illustrates why it is necessary to define the product of two ideals to be the ideal generated by $\{ab : a \in I, b\in J\}$.
    3. Give an example of a ring of integers $\O _K$ of a number field, and ideals $I$ and $J$ such that $\{ab : a \in I, b\in J\}$ is not an ideal.

11. 1. Let $k$ be a field. Prove that $k[x]$ is a Dedekind domain.
    2. (Problem 1.12 from Swinnerton-Dyer) Let $x$ be an indeterminate. Show that the ring $\mathbf{Z}[x]$ is Noetherian and integrally closed in its field of fractions, but is not a Dedekind domain.

12. Use to write each of the following (fractional) ideals as a product of explicitly given prime ideals:
    1. The ideal $(2004)$ in $\mathbf{Q}(\sqrt{-1})$.
    2. The ideals $I=(7)$ and $J=(3)$ in the ring of integers of $\mathbf{Q}(\zeta_7)$, where $\zeta_7$ is a root of the irreducible polynomial $x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$. (The field $\mathbf{Q}(\zeta_7)$ is called the $7$th cyclotomic field.)
    3. The principal fractional ideal $(3/8)$ in $\mathbf{Q}(\sqrt{5})$.

13. Suppose $R$ is an order in the ring $\O _K$ of integers of a number field. (Recall that an order is a subring of finite index in $\O _K$.) For each of the following questions, either explain why the answer is yes for any possible order $R$ in any $\O _K$, or find one specific counterexample:
    1. Is $R$ necessarily Noetherian?
    2. Is $R$ necessarily integrally closed in its field of fractions?
    3. Is every nonzero prime ideal of $R$ necessarily maximal?
    4. Is it always possible to write every ideal of $R$ uniquely as a product of prime ideals of $R$?

14. Let $\O _K$ be the ring of integers of a number field $K$. Prove that the group of fractional ideals of $\O _K$, under multiplication is (non-canonically) isomorphic to the group of positive rational numbers under multiplication.

15. 1. Suppose $K$ is a number field of degree $2$. Prove that $\O _K=\mathbf{Z}[a]$ for some $a\in\O _K$.
    2. Prove that if $K$ and $K'$ are two number fields of degree $2$ and $\Disc (\O _K) = \Disc (\O _{K'})$ then $K=K'$.

16. (*) Does there exist a number field $K$ of degree $4$ such that $\O _K\neq \mathbf{Z}[a]$ for all $a\in\O _K$? If so, give an explicit example.

17. Let $K$ be the quintic number field generated by a root of $x^5+7x^4+3x^2-x+1$. Draw a diagram (be creative) that illustrates the factorization of every prime $p\in\mathbf{Z}$, with $p<100$, in $\O _K$.

18. (Problem 1.9 in Swinnerton-Dyer) Show that the only solutions $x,y\in\mathbf{Z}$ to $y^2 = x^3 - 13$ are given by $x=17, y=\pm 70$, as follows. Factor the equation $y^2+13=x^3$ in the number field $\mathbf{Q}(\sqrt{-13})$, which has class number $2$. Show that if $x, y$ is an integer solution then the ideal $(y+\sqrt{-13})$ must be the cube of an ideal, and hence $y+\sqrt{-13} = (a+b\sqrt{-13})^3$; thus $1=b(3a^2-13b^2)$.

19. Suppose $I$ and $J$ are ideals in the ring $\O _K$ of integers of a number field $K$. Does $IJ=I\cap J$? Prove or give a counterexample.

20. Let $\O _K$ be the ring of integers $\mathbf{Q}(\sqrt{5})$, and let
    $$I=(5, 2+\sqrt{5})$$   and$$\qquad J=(209,(389+\sqrt{5})/2)$$
    be integral ideals of $\O _K$.
    1. Find an element of $\O _K$ that is congruent to $\sqrt{5}$ modulo $I$ and is congruent to $1-\sqrt{5}$ modulo $J$.
    2. What is the cardinality of $(\O _K/I) \oplus (\O _K/J)$?
    3. Find an element $a\in I$ such that $(a)/I$ is coprime to $J$.

21. Let $\O _K$ be the ring of integers of a number field $K$, and suppose $K$ has exactly $2s$ complex embeddings. Prove that the sign of $\Disc (\O _K)$ is $(-1)^s$.

22. (*) Suppose $\O$ is an order in the ring of integers $\O _K$ of a number field. Is every ideal in $\O$ necessarily generated by two elements? (Answer: No. Challenge: Given an example.)

23. Find representative ideals for each element of the class group of $\mathbf{Q}(\sqrt{-23})$. Illustrate how to use the Minkowski bound to prove that your list of representatives is complete.

24. Suppose $\O$ is an order in the ring of integers $\O _K$ of a number field. Is every ideal in $\O$ necessarily generated by two elements?

25. Let $K$ be a number field of degree $n>1$ with $s$ pairs of complex conjugate embeddings. Prove that
    $$\left(\frac{\pi}{4}\right)^s\cdot \frac{n^n}{n!} > 1.$$

26. Do the exercise on page 19 of Swinnerton-Dyer, which shows that the quantity $C_{r,s}$ in the finiteness of class group theorem can be taken to be $\left(\frac{4}{\pi}\right)^{s} \frac{n!}{n^n}$.

27. Let $\alpha$ denote a root of $x^3-x+2$ and let $K=\mathbf{Q}(\alpha)$. Show that $\O _K=\mathbf{Z}[\alpha]$ and that $K$ has class number $1$ (don't just read this off from the output of the commands MaximalOrder and ClassNumber). [Hint: consider the square factors of the discriminant of $x^3-x+2$ and show that $\frac{1}{2}(a+b\alpha + c\alpha^2)$ is an algebra integer if and only if $a$, $b$, and $c$ are all even.]

28. If $S$ is a closed, bounded, convex, symmetric set in $\mathbf{R}^n$ with $\Vol (S)\geq m 2^n$, for some positive integer $m$, show that $S$ contains at least $2m$ nonzero points in $\mathbf{Z}^n$.

29. Prove that any finite subgroup of the multiplicative group of a field is cyclic.

30. For a given number field $K$, which seems more difficult for to compute, the class groups or explicit generators for the group of units? It is very difficult (but not impossible) to not get full credit on this problem. Play around with some examples, see what seems more difficult, and justify your response with examples. (This problem might be annoying to do using the web page, since it kills your job after 30 seconds. Feel free to request a binary of from me, or an account on MECCAH (Mathematics Extreme Computation Cluster at Harvard).)

31. 1. Prove that there is no number field $K$ such that $U_K\cong \mathbf{Z}/10\mathbf{Z}$.
    2. Is there a number field $K$ such that $U_K\cong \mathbf{Z}\times \mathbf{Z}/6\mathbf{Z}$?

32. Prove that the rank of $U_K$ is unbounded as $K$ varies over all number fields.

33. Let $K=\mathbf{Q}(\zeta_5)$.
    1. Show that $r=0$ and $s=2$.
    2. Find explicitly generators for the group of units of $U_K$ (you can use for this).
    3. Draw an illustration of the log map $\varphi :U_K\to \mathbf{R}^2$, including the hyperplane $x_1+x_2=0$ and the lattice in the hyperplane spanned by the image of $U_K$.

34. Find the group of units of $\mathbf{Q}(\zeta_n)$ as an abstract group as a function of $n$. (I.e., find the number of cyclic factors and the size of the torsion subgroup. You do not have to find explicit generators!)

35. Let $K=\mathbf{Q}(a)$, where $a$ is a root $x^3-3x+1$.
    1. Show that $r=3$.
    2. Find explicitly the log embedding of $U_K$ into a $2$-dimensional hyperplane in $\mathbf{R}^3$, and draw a picture.

36. Prove that if $K$ is a quadratic field and the torsion subgroup of $U_K$ has order bigger than $2$, then $K=\mathbf{Q}(\sqrt{-3})$ or $K=\mathbf{Q}(\sqrt{-1})$.

37. A is a real algebraic integer, greater than 1, with the property that all of its conjugates lie on or within the unit circle, and at least one conjugate lies on the unit circle. By any method (including ``google''), give two examples of Salem numbers.

38. Let $p\in\mathbf{Z}$ and let $K$ be a number field. Show that $\Norm _{K/\mathbf{Q}}(p\O _K) = p^{[K:\mathbf{Q}]}$.

39. A totally real number field is a number field in which all embeddings into $\mathbf{C}$ have image in $\mathbf{R}$. Prove there are totally real number fields of degree $p$, for every prime $p$. [Hint: Let $\zeta_n$ denote a primitive $n$th root of unity. For $n\geq 3$, show that $\mathbf{Q}(\zeta_n+1/\zeta_n)$ is totally real of degree $\varphi (n)/2$. Now prove that $\varphi (n)/2$ can be made divisible by any prime.]

40. Give an example of a number field $K/\mathbf{Q}$ and a prime $p$ such that the $e_i$ in the factorization of $p\O _K$ are not all the same.

41. Let $K$ be a number field. Give the ``simplest'' proof you can think of that there are only finitely many primes that ramify (i.e., have some $e_i>1$) in $K$. [The meaning of ``simplest'' is a matter of taste.]

42. Give examples to show that for $K/\mathbf{Q}$ a Galois extension, the quantity $e$ can be arbirarily large and $f$ can be arbitrarily large.

43. Suppose $K/\mathbf{Q}$ is Galois and $p$ is a prime such that $p\O _K$ is also prime (i.e., $p$ is inert in $K$). Show that $\Gal (K/\mathbf{Q})$ is a cyclic group.

44. (Problem 7, page 116, from Marcus Number Fields) For each of the following, find a prime $p$ and quadratic extensions $K$ and $L$ of $\mathbf{Q}$ that illustrates the assertion:
    1. The prime $p$ can be totally ramified in $K$ and $L$ without being totally ramified in $KL$.
    2. The fields $K$ and $L$ can each contain unique primes lying over $p$ while $KL$ does not.
    3. The prime $p$ can be inert in $K$ and $L$ without being inert in $KL$.
    4. The residue field extensions of $\mathbf{F}_p$ can be trivial for $K$ and $L$ without being trivial for $KL$.

45. Let $S_3$ by the symmetric group on three symbols, which has order $6$.
    1. Observe that $S_3\cong D_3$, where $D_3$ is the dihedral group of order $6$, which is the group of symmetries of an equilateral triangle.
    2. Use (45a) to write down an explicit embedding $S_3\hookrightarrow \GL _2(\mathbf{C})$.
    3. Let $K$ be the number field $\mathbf{Q}(\sqrt[3]{2},\omega)$, where $\omega^3=1$ is a nontrivial cube root of unity. Show that $K$ is a Galois extension with Galois group isomorphic to $S_3$.
    4. We thus obtain a $2$-dimensional irreducible complex Galois representation
       $$\rho:\Gal (\overline{\mathbf{Q}}/\mathbf{Q}) \to \Gal (K/\mathbf{Q})\cong S_3 \subset \GL _2(\mathbf{C}).$$
       Compute a representative matrix of $\Frob _p$ and the characteristic polynomial of $\Frob _p$ for $p=5,7,11,13$.

46. Let $K=\mathbf{Q}(\sqrt{2},\sqrt{3},\sqrt{5},\sqrt{7})$. Show that $K$ is Galois over $\mathbf{Q}$, compute the Galois group of $K$, and compute $\Frob _{37}$.

47. Let $k$ be any field. Prove that the only nontrivial valuations on $k(t)$ which are trivial on $k$ are equivalent to the valuation (15.3.3) or (15.3.4) of page .
48. A field with the topology induced by a valuation is a topological field, i.e., the operations sum, product, and reciprocal are continuous.
49. Give an example of a non-archimedean valuation on a field that is not discrete.
50. Prove that the field $\mathbf{Q}_p$ of $p$-adic numbers is uncountable.
51. Prove that the polynomial $f(x)=x^3 - 3x^2 + 2x + 5$ has all its roots in $\mathbf{Q}_5$, and find the $5$-adic valuations of each of these roots. (You might need to use Hensel's lemma, which we don't discuss in detail in this book. See [Cas67, App. C].)

52. In this problem you will compute an example of weak approximation, like I did in the Example 16.3.3. Let $K=\mathbf{Q}$, let $\left\vert \cdot \right\vert _7$ be the $7$-adic absolute value, let $\left\vert \cdot \right\vert _{11}$ be the $11$-adic absolute value, and let $\left\vert \cdot \right\vert _{\infty}$ be the usual archimedean absolute value. Find an element $b\in \mathbf{Q}$ such that $\left\vert b-a_i\right\vert _i<\frac{1}{10}$, where $a_7 = 1$, $a_{11} = 2$, and $a_{\infty} = -2004$.

53. Prove that $-9$ has a cube root in $\mathbf{Q}_{10}$ using the following strategy (this is a special case of Hensel's Lemma, which you can read about in an appendix to Cassel's article).
    1. Show that there is an element $\alpha\in\mathbf{Z}$ such that $\alpha^3\equiv 9\pmod{10^3}$.
    2. Suppose $n\geq 3$. Use induction to show that if $\alpha_1\in\mathbf{Z}$ and $\alpha^3\equiv 9\pmod{10^n}$, then there exists $\alpha_2\in\mathbf{Z}$ such that $\alpha_2^3\equiv 9\pmod{10^{n+1}}$. (Hint: Show that there is an integer $b$ such that $(\alpha_1 + b\cdot 10^{n})^3 \equiv 9\pmod{10^{n+1}}$.)
    3. Conclude that $9$ has a cube root in $\mathbf{Q}_{10}$.

54. Compute the first $5$ digits of the $10$-adic expansions of the following rational numbers:
    $$\frac{13}{2}, \quad \frac{1}{389}, \quad \frac{17}{19},$$    the 4 square roots of $41$$$.$$

55. Let $N>1$ be an integer. Prove that the series
    $$\sum_{n=1}^{\infty} (-1)^{n+1}n! = 1! - 2! + 3! - 4! + 5! - 6! + \cdots.$$
    converges in $\mathbf{Q}_N$.

56. Prove that $-9$ has a cube root in $\mathbf{Q}_{10}$ using the following strategy (this is a special case of ``Hensel's Lemma'').
    1. Show that there is $\alpha\in\mathbf{Z}$ such that $\alpha^3\equiv 9\pmod{10^3}$.
    2. Suppose $n\geq 3$. Use induction to show that if $\alpha_1\in\mathbf{Z}$ and $\alpha^3\equiv 9\pmod{10^n}$, then there exists $\alpha_2\in\mathbf{Z}$ such that $\alpha_2^3\equiv 9\pmod{10^{n+1}}$. (Hint: Show that there is an integer $b$ such that $(\alpha_1 + b10^{n})^3 \equiv 9\pmod{10^{n+1}}$.)
    3. Conclude that $9$ has a cube root in $\mathbf{Q}_{10}$.

57. Let $N>1$ be an integer.
    1. Prove that $\mathbf{Q}_N$ is equipped with a natural ring structure.
    2. If $N$ is prime, prove that $\mathbf{Q}_N$ is a field.

58. 1. Let $p$ and $q$ be distinct primes. Prove that $\mathbf{Q}_{pq} \cong \mathbf{Q}_p \times \mathbf{Q}_q$.
    2. Is $\mathbf{Q}_{p^2}$ isomorphic to either of $\mathbf{Q}_p\times \mathbf{Q}_p$ or $\mathbf{Q}_p$?

59. Prove that every finite extension of $\mathbf{Q}_p$ ``comes from'' an extension of  $\mathbf{Q}$, in the following sense. Given an irreducible polynomial $f\in\mathbf{Q}_p[x]$ there exists an irreducible polynomial $g\in \mathbf{Q}[x]$ such that the fields $\mathbf{Q}_p[x]/(f)$ and $\mathbf{Q}_p[x]/(g)$ are isomorphic. [Hint: Choose each coefficient of $g$ to be sufficiently close to the corresponding coefficient of $f$, then use Hensel's lemma to show that $g$ has a root in $\mathbf{Q}_p[x]/(f)$.]

60. Find the $3$-adic expansion to precision 4 of each root of the following polynomial over $\mathbf{Q}_3$:
    $$f = x^3 - 3x^2 + 2x + 3 \in \mathbf{Q}_3[x].$$
    Your solution should conclude with three expressions of the form
    $$a_0 + a_1\cdot 3 + a_2\cdot 3^2 + a_3 \cdot 3^3 + O(3^4).$$

61. 1. Find the normalized Haar measure of the following subset of $\mathbf{Q}_7^+$:
       $$U = B\left(28,\frac{1}{50}\right) = \left\lbrace x\in \mathbf{Q}_7 : \left\vert x-28\right\vert < \frac{1}{50}\right\rbrace.$$

    2. Find the normalized Haar measure of the subset $\mathbf{Z}_7^*$ of $\mathbf{Q}_7^*$.

62. Suppose that $K$ is a finite extension of $\mathbf{Q}_p$ and $L$ is a finite extension of $\mathbf{Q}_q$, with $p\neq q$ and assume that $K$ and $L$ have the same degree. Prove that there is a polynomial $g\in \mathbf{Q}[x]$ such that $\mathbf{Q}_p[x]/(g)\cong K$ and $\mathbf{Q}_q[x]/(g)\cong L$. [Hint: Combine your solution to 59 with the weak approximation theorem.]

63. Prove that the ring $C$ defined in Section 9 really is the tensor product of $A$ and $B$, i.e., that it satisfies the defining universal mapping property for tensor products. Part of this problem is for you to look up a functorial definition of tensor product.

64. Find a zero divisor pair in $\mathbf{Q}(\sqrt{5})\otimes _\mathbf{Q}\mathbf{Q}(\sqrt{5})$.

65. 1. Is $\mathbf{Q}(\sqrt{5})\otimes _\mathbf{Q}\mathbf{Q}(\sqrt{-5})$ a field?
    2. Is $\mathbf{Q}(\sqrt[4]{5})\otimes _\mathbf{Q}\mathbf{Q}(\sqrt[4]{-5})\otimes _\mathbf{Q}\mathbf{Q}(\sqrt{-1})$ a field?

66. Suppose $\zeta_5$ denotes a primitive $5$th root of unity. For any prime $p$, consider the tensor product $\mathbf{Q}_p \otimes _\mathbf{Q} \mathbf{Q}(\zeta_5) = K_1\oplus \cdots \oplus K_{n(p)}$. Find a simple formula for the number $n(p)$ of fields appearing in the decomposition of the tensor product $\mathbf{Q}_p \otimes _\mathbf{Q}\mathbf{Q}(\zeta_5)$. To get full credit on this problem your formula must be correct, but you do not have to prove that it is correct.

67. Suppose $\left\vert \cdot \right\vert _1$ and $\left\vert \cdot \right\vert _2$ are equivalent norms on a finite-dimensional vector space $V$ over a field $K$ (with valuation $\left\vert \cdot \right\vert$). Carefully prove that the topology induced by $\left\vert \cdot \right\vert _1$ is the same as that induced by $\left\vert \cdot \right\vert _2$.

68. Suppose $K$ and $L$ are number fields (i.e., finite extensions of $\mathbf{Q}$). Is it possible for the tensor product $K\otimes _\mathbf{Q}L$ to contain a nilpotent element? (A nonzero element $a$ in a ring $R$ is if there exists $n>1$ such that $a^n=0$.)

69. Let $K$ be the number field $\mathbf{Q}(\sqrt[5]{2})$.
    1. In how many ways does the $2$-adic valuation $\left\vert \cdot \right\vert _2$ on $\mathbf{Q}$ extend to a valuation on $K$?
    2. Let $v=\left\vert \cdot \right\vert$ be a valuation on $K$ that extends $\left\vert \cdot \right\vert _2$. Let $K_v$ be the completion of $K$ with respect to $v$. What is the residue class field $\mathbf{F}$ of $K_v$?

70. Prove that the product formula holds for $\mathbf{F}(t)$ similar to the proof we gave in class using Ostrowski's theorem for $\mathbf{Q}$. You may use the analogue of Ostrowski's theorem for $\mathbf{F}(t)$, which you had on a previous homework assignment. (Don't give a measure-theoretic proof.)
71. Prove Theorem 20.3.5, that ``The global field $K$ is discrete in $\AA _K$ and the quotient $\AA _K^+/K^+$ of additive groups is compact in the quotient topology.'' in the case when $K$ is a finite extension of $\mathbf{F}(t)$, where $\mathbf{F}$ is a finite field.