Exercises

  1. Let $ A=\left(
\begin{matrix}1&2&3 4&5&6 7&8&9
\end{matrix}\right)$.
    1. Find the Smith normal form of $ A$.
    2. Prove that the cokernel of the map $ \mathbf{Z}^3\to \mathbf{Z}^3$ given by multiplication by $ A$ is isomorphic to $ \mathbf{Z}/3\mathbf{Z}\oplus \mathbf{Z}$.

  2. Show that the minimal polynomial of an algebraic number $ \alpha\in\overline{\mathbf{Q}}$ is unique.

  3. Which of the following rings have infinitely many prime ideals?
    1. The integers $ \mathbf {Z}$.
    2. The ring $ \mathbf{Z}[x]$ of polynomials over $ \mathbf {Z}$.
    3. The quotient ring $ \mathbf{C}[x]/(x^{2005}-1)$.
    4. The ring $ (\mathbf{Z}/6\mathbf{Z})[x]$ of polynomials over the ring $ \mathbf{Z}/6\mathbf{Z}$.
    5. The quotient ring $ \mathbf{Z}/n\mathbf{Z}$, for a fixed positive integer $ n$.
    6. The rational numbers  $ \mathbf {Q}$.
    7. The polynomial ring $ \mathbf{Q}[x,y,z]$ in three variables.

  4. Which of the following numbers are algebraic integers?
    1. The number $ (1+\sqrt{5})/2$.
    2. The number $ (2+\sqrt{5})/2$.
    3. The value of the infinite sum $ \sum_{n=1}^{\infty} 1/n^2$.
    4. The number $ \alpha/3$, where $ \alpha$ is a root of $ x^4 + 54x + 243$.

  5. Prove that $ \overline{\mathbf{Z}}$ is not noetherian.

  6. Let $ \alpha = \sqrt{2} + \frac{1+\sqrt{5}}{2}$.
    1. Is $ \alpha$ an algebraic integer?
    2. Explicitly write down the minimal polynomial of $ \alpha$ as an element of $ \mathbf{Q}[x]$.

  7. Which are the following rings are orders in the given number field.
    1. The ring $ R = \mathbf{Z}[i]$ in the number field $ \mathbf{Q}(i)$.
    2. The ring $ R = \mathbf{Z}[i/2]$ in the number field $ \mathbf{Q}(i)$.
    3. The ring $ R = \mathbf{Z}[17i]$ in the number field $ \mathbf{Q}(i)$.
    4. The ring $ R = \mathbf{Z}[i]$ in the number field $ \mathbf{Q}(\sqrt[4]{-1})$.

  8. We showed in the text (see Proposition 3.1.3) that $ \overline{\mathbf{Z}}$ is integrally closed in its field of fractions. Prove that and every nonzero prime ideal of $ \overline{\mathbf{Z}}$ is maximal. Thus $ \overline{\mathbf{Z}}$ is not a Dedekind domain only because it is not noetherian.

  9. Let $ K$ be a field.
    1. Prove that the polynomial ring $ K[x]$ is a Dedekind domain.
    2. Is $ \mathbf{Z}[x]$ a Dedekind domain?

  10. Prove that every finite integral domain is a field.

    1. Give an example of two ideals $ I,J$ in a commutative ring $ R$ whose product is not equal to the set $ \{ab : a \in I, b \in J\}$.
    2. Suppose $ R$ is a principal ideal domain. Is it always the case that

      $\displaystyle IJ = \{ab : a \in I, b \in J\}
$

      for all ideals $ I,J$ in $ R$?

  11. Is the set $ \mathbf{Z}[\frac{1}{2}]$ of rational numbers with denominator a power of $ 2$ a fractional ideal?

  12. Suppose you had the choice of the following two jobs20.1:
    Job 1
    Starting with an annual salary of $1000, and a $200 increase every year.
    Job 2
    Starting with a semiannual salary of $500, and an increase of $50 every 6 months.
    In all other respects, the two jobs are exactly alike. Which is the better offer (after the first year)? Write a Sage program that creates a table showing how much money you will receive at the end of each year for each job. (Of course you could easily do this by hand - the point is to get familiar with Sage.)

  13. Let $ \O_K$ be the ring of integers of a number field. Let $ F_K$ denote the abelian group of fractional ideals of $ \O_K$.
    1. Prove that $ F_K$ is torsion free.
    2. Prove that $ F_K$ is not finitely generated.
    3. Prove that $ F_K$ is countable.
    4. Conclude that if $ K$ and $ L$ are number fields, then there exists some (non-canonical) isomorphism of groups $ F_K\approx F_L$.

  14. From basic definitions, find the rings of integers of the fields $ \mathbf{Q}(\sqrt{11})$ and $ \mathbf{Q}(\sqrt{-6})$.

  15. In this problem, you will give an example to illustrate the failure of unique factorization in the ring $ \O_K$ of integers of $ \mathbf{Q}(\sqrt{-6})$.
    1. Give an element $ \alpha\in
\O_K$ that factors in two distinct ways into irreducible elements.
    2. Observe explicitly that the $ (\alpha)$ factors uniquely, i.e., the two distinct factorization in the previous part of this problem do not lead to two distinct factorization of the ideal $ (\alpha)$ into prime ideals.

  16. Factor the ideal $ (10)$ as a product of primes in the ring of integers of $ \mathbf{Q}(\sqrt{11})$. You're allowed to use a computer, as long as you show the commands you use.

  17. Let $ \O_K$ be the ring of integers of a number field $ K$, and let $ p\in\mathbf{Z}$ be a prime number. What is the cardinality of $ \O_K/(p)$ in terms of $ p$ and $ [K:\mathbf{Q}]$, where $ (p)$ is the ideal of $ \O_K$ generated by $ p$?

  18. Give an example of each of the following, with proof:
    1. A non-principal ideal in a ring.
    2. A module that is not finitely generated.
    3. The ring of integers of a number field of degree $ 3$.
    4. An order in the ring of integers of a number field of degree $ 5$.
    5. The matrix on $ K$ of left multiplication by an element of $ K$, where $ K$ is a degree $ 3$ number field.
    6. An integral domain that is not integrally closed in its field of fractions.
    7. A Dedekind domain with finite cardinality.
    8. A fractional ideal of the ring of integers of a number field that is not an integral ideal.

  19. Let $ \varphi :R\to S$ be a homomorphism of (commutative) rings.
    1. Prove that if $ I\subset S$ is an ideal, then $ \varphi ^{-1}(I)$ is an ideal of $ R$.
    2. Prove moreover that if $ I$ is prime, then $ \varphi ^{-1}(I)$ is also prime.

  20. Let $ \O_K$ be the ring of integers of a number field. The Zariski topology on the set $ X=\Spec (\O_K)$ of all prime ideals of $ \O_K$ has closed sets the sets of the form

    $\displaystyle V(I) = \{ \mathfrak{p}\in X : \mathfrak{p}\mid I\},
$

    where $ I$ varies through all ideals of $ \O_K$, and $ \mathfrak{p}\mid I$ means that $ I\subset \mathfrak{p}$.
    1. Prove that the collection of closed sets of the form $ V(I)$ is a topology on $ X$.
    2. Let $ Y$ be the subset of nonzero prime ideals of $ \O_K$, with the induced topology. Use unique factorization of ideals to prove that the closed subsets of $ Y$ are exactly the finite subsets of $ Y$ along with the set $ Y$.
    3. Prove that the conclusion of (a) is still true if $ \O_K$ is replaced by an order in $ \O_K$, i.e., a subring that has finite index in $ \O_K$ as a $ \mathbf {Z}$-module.

  21. Explicitly factor the ideals generated by each of $ 2$, $ 3$, and $ 5$ in the ring of integers of $ \mathbf{Q}(\sqrt[3]{2})$. (Thus you'll factor $ 3$ separate ideals as products of prime ideals.) You may assume that the ring of integers of $ \mathbf{Q}(\sqrt[3]{2})$ is $ \mathbf{Z}[\sqrt[3]{2}]$, but do not simply use a computer command to do the factorizations.

  22. Let $ K=\mathbf{Q}(\zeta_{13})$,where $ \zeta_{13}$ is a primitive $ 13$th root of unity. Note that $ K$ has ring of integers $ \O_K=\mathbf{Z}[\zeta_{13}]$.
    1. Factor $ 2$, $ 3$, $ 5$, $ 7$, $ 11$, and $ 13$ in the ring of integers $ \O_K$. You may use a computer.
    2. For $ p\neq 13$, find a conjectural relationship between the number of prime ideal factors of $ p\O_K$ and the order of the reduction of $ p$ in $ (\mathbf{Z}/13\mathbf{Z})^*$.
    3. Compute the minimal polynomial $ f(x) \in \mathbf{Z}[x]$ of $ \zeta_{13}$. Reinterpret your conjecture as a conjecture that relates the degrees of the irreducible factors of $ f(x)\pmod{p}$ to the order of $ p$ modulo $ 13$. Does your conjecture remind you of quadratic reciprocity?

    1. Find by hand and with proof the ring of integers of each of the following two fields: $ \mathbf{Q}(\sqrt{5})$, $ \mathbf{Q}(i)$.
    2. Find the ring of integers of $ \mathbf{Q}(a)$, where $ a^5+7a+1=0$ using a computer.

  23. Let $ p$ be a prime. Let $ \O_K$ be the ring of integers of a number field $ K$, and suppose $ a\in\O_K$ is such that $ [\O_K:\mathbf{Z}[a]]$ is finite and coprime to $ p$. Let $ f(x)$ be the minimal polynomial of $ a$. We proved in class that if the reduction $ \overline{f}\in\mathbf{F}_p[x]$ of $ f$ factors as

    $\displaystyle \overline{f} = \prod g_i^{e_i},
$

    where the $ g_i$ are distinct irreducible polynomials in $ \mathbf{F}_p[x]$, then the primes appearing in the factorization of $ p\O_K$ are the ideals $ (p,g_i(a))$. In class, we did not prove that the exponents of these primes in the factorization of $ p\O_K$ are the $ e_i$. Prove this.

  24. Let $ a_1 = 1+i$, $ a_2 = 3+2i$, and $ a_3 = 3+4i$ as elements of $ \mathbf{Z}[i]$.
    1. Prove that the ideals $ I_1=(a_1)$, $ I_2=(a_2)$, and $ I_3=(a_3)$ are coprime in pairs.
    2. Compute $ \char93 \mathbf{Z}[i]/(I_1 I_2 I_3)$.
    3. Find a single element in $ \mathbf{Z}[i]$ that is congruent to $ n$ modulo $ I_n$, for each $ n\leq 3$.

  25. Find an example of a field $ K$ of degree at least $ 4$ such that the ring $ \O_K$ of integers of $ K$ is not of the form $ \mathbf{Z}[a]$ for any $ a\in\O_K$.

  26. Let $ \mathfrak{p}$ be a prime ideal of $ \O_K$, and suppose that $ \O_K/\mathfrak{p}$ is a finite field of characteristic $ p\in\mathbf{Z}$. Prove that there is an element $ \alpha\in
\O_K$ such that $ \mathfrak{p}=(p,\alpha)$. This justifies why we can represent prime ideals of $ \O_K$ as pairs $ (p,\alpha)$, as is done in SAGE. (More generally, if $ I$ is an ideal of $ \O_K$, we can choose one of the elements of $ I$ to be any nonzero element of $ I$.)

  27. (*) Give an example of an order $ \O$ in the ring of integers of a number field and an ideal $ I$ such that $ I$ cannot be generated by $ 2$ elements as an ideal. Does the Chinese Remainder Theorem hold in $ \O$? [The (*) means that this problem is more difficult than usual.]

  28. For each of the following three fields, determining if there is an order of discriminant $ 20$ contained in its ring of integers:

    $\displaystyle K = \mathbf{Q}(\sqrt{5}), \quad K=\mathbf{Q}(\sqrt[3]{2}),$   and$\displaystyle \ldots
$

    $ K$ any extension of $ \mathbf {Q}$ of degree $ 2005$. [Hint: for the last one, apply the exact form of our theorem about finiteness of class groups to the unit ideal to show that the discriminant of a degree $ 2005$ field must be large.]

  29. Prove that the quantity $ C_{r,s}$ in our theorem about finiteness of the class group can be taken to be $ \left(\frac{4}{\pi}\right)^{s} \frac{n!}{n^n}$, as follows (adapted from [SD01, pg. 19]): Let $ S$ be the set of elements $ (x_1,\ldots, x_{n})\in\mathbf{R}^n$ such that

    $\displaystyle \vert x_1\vert + \cdots \vert x_{r}\vert + 2 \sum_{v=r+1}^{r+s}
\sqrt{x_v^2 + x_{v+s}^2} \leq 1.
$

    1. Prove that $ S$ is convex and that $ M=n^{-n}$, where

      $\displaystyle M = \max\{ \vert x_1\cdots x_r\cdot (x_{r+1}^2 + x_{(r+1)+s}^2)\cdots (x_{r+s}^2 + x_n^2)\vert : (x_1,\ldots, x_n) \in S\}.
$

      [Hint: For convexity, use the triangle inequality and that for $ 0\leq \lambda \leq 1$, we have

      $\displaystyle \lambda\sqrt{x_1^2 + y_1^2}$ $\displaystyle + (1-\lambda)\sqrt{x_2^2+y_2^2}$    
        $\displaystyle \geq\sqrt{(\lambda x_1 + (1-\lambda)x_2)^2 + (\lambda y_1 + (1-\lambda)y_2)^2}$    

      for $ 0\leq \lambda \leq 1$. In polar coordinates this last inequality is

      $\displaystyle \lambda r_1 + (1-\lambda)r_2 \geq
\sqrt{\lambda^2 r_1^2 + 2\lambda(1-\lambda) r_1 r_2 \cos(\theta_1 - \theta_2) + (1-\lambda)^2 r_2^2},
$

      which is trivial. That $ M\leq n^{-n}$ follows from the inequality between the arithmetic and geometric means.
    2. Transforming pairs $ x_v, x_{v+s}$ from Cartesian to polar coordinates, show also that $ v=2^{r}(2\pi)^s D_{r,s}(1)$, where

      $\displaystyle D_{\ell,m}(t) = \int \cdots \int_{\mathcal{R}_{\ell,m}(t)}
y_1 \cdots y_m dx_1 \cdots dx_{\ell} dy_1 \cdots dy_m
$

      and $ \mathcal{R_{\ell,m}}(t)$ is given by $ x_{\rho}\geq 0$ ( $ 1\leq \rho\leq \ell$), $ y_{\rho}\geq 0$ ( $ 1\leq \rho\leq m$) and

      $\displaystyle x_1 + \cdots + x_{\ell} + 2(y_1+\cdots +y_m) \leq t.
$

    3. Prove that

      $\displaystyle D_{\ell,m}(t) = \int_{0}^t D_{\ell-1,m}(t-x)dx
=\int_{0}^{t/2} D_{\ell,m-1}(t-2y)y dy
$

      and deduce by induction that

      $\displaystyle D_{\ell,m}(t) = \frac{4^{-m}t^{\ell+2m}}{(\ell+2m)!}
$

  30. Let $ K$ vary through all number fields. What torsion subgroups $ (U_K)_{\tor }$ actually occur?

  31. If $ U_K \approx \mathbf{Z}^n \times (U_K)_{\tor }$, we say that $ U_K$ has rank $ n$. Let $ K$ vary through all number fields. What ranks actually occur?

  32. Let $ K$ vary through all number fields such that the group $ U_K$ of units of $ K$ is a finite group. What finite groups $ U_K$ actually occur?

  33. Let $ K=\mathbf{Q}(\zeta_5)$.
    1. Show that $ r=0$ and $ s=2$.
    2. Find explicit generators for the group of units $ U_K$.
    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. Let $ K$ be a number field. Prove that $ p\mid d_K$ if and only if $ p$ ramifies in $ K$. (Note: This fact is proved in many books.)

  35. Using Zorn's lemma, show that there are homomorphisms $ {\mathrm{Gal}}(\overline{\mathbf{Q}}/\mathbf{Q})\to\{\pm 1\}$ with finite image that are not continuous, since they do not factor through the Galois group of any finite Galois extension. [Hint: The extension $ \mathbf{Q}(\sqrt{d}, d \in \mathbf{Q}^*/(\mathbf{Q}^*)^2)$ is an extension of  $ \mathbf {Q}$ with Galois group $ X\approx \prod \mathbf{F}_2$. The index-two open subgroups of $ X$ correspond to the quadratic extensions of  $ \mathbf {Q}$. However, Zorn's lemma implies that $ X$ contains many index-two subgroups that do not correspond to quadratic extensions of  $ \mathbf {Q}$.]

    1. Give an example of a finite nontrivial Galois extension $ K$ of $ \mathbf {Q}$ and a prime ideal $ \mathfrak{p}$ such that $ D_\mathfrak{p}= \Gal (K/\mathbf{Q})$.
    2. Give an example of a finite nontrivial Galois extension $ K$ of $ \mathbf {Q}$ and a prime ideal $ \mathfrak{p}$ such that $ D_\mathfrak{p}$ has order $ 1$.
    3. Give an example of a finite Galois extension $ K$ of $ \mathbf {Q}$ and a prime ideal $ \mathfrak{p}$ such that $ D_\mathfrak{p}$ is not a normal subgroup of $ \Gal (K/\mathbf{Q})$.
    4. Give an example of a finite Galois extension $ K$ of $ \mathbf {Q}$ and a prime ideal $ \mathfrak{p}$ such that $ I_\mathfrak{p}$ is not a normal subgroup of $ \Gal (K/\mathbf{Q})$.

  36. 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 (39a) 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

      $\displaystyle \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$.

  37. Look up the Riemann-Roch theorem in a book on algebraic curves.
    1. Write it down in your own words.
    2. Let $ E$ be an elliptic curve over a field $ K$. Use the Riemann-Roch theorem to deduce that the natural map

      $\displaystyle E(K) \to \Pic ^0(E/K)$

      is an isomorphism.

  38. Suppose $ G$ is a finite group and $ A$ is a finite $ G$-module. Prove that for any $ q$, the group $ \H^q(G,A)$ is a torsion abelian group of exponent dividing the order $ \char93 A$ of $ A$.

  39. Let $ K=\mathbf{Q}(\sqrt{5})$ and let $ A=U_K$ be the group of units of $ K$, which is a module over the group $ G=\Gal (K/\mathbf{Q})$. Compute the cohomology groups $ \H^0(G,A)$ and $ \H^1(G,A)$. (You shouldn't use a computer, except maybe to determine $ U_K$.)

  40. Let $ K=\mathbf{Q}(\sqrt{-23})$ and let $ C$ be the class group of $ \mathbf{Q}(\sqrt{-23})$, which is a module over the Galois group $ G=\Gal (K/\mathbf{Q})$. Determine $ \H^0(G,C)$ and $ \H^1(G,C)$.

  41. Let $ E$ be the elliptic curve $ y^2=x^3+x+1$. Let $ E[2]$ be the group of points of order dividing $ 2$ on $ E$. Let

    $\displaystyle \overline{\rho}_{E,2}:\Gal (\overline{\mathbf{Q}}/\mathbf{Q}) \to \Aut (E[2])
$

    be the mod $ 2$ Galois representation associated to $ E$.
    1. Find the fixed field $ K$ of $ \ker(\overline{\rho}_{E,2})$.
    2. Is $ \overline{\rho}_{E,2}$ surjective?
    3. Find the group $ \Gal (K/\mathbf{Q})$.
    4. Which primes are ramified in $ K$?
    5. Let $ I$ be an inertia group above $ 2$, which is one of the ramified primes. Determine $ E[2]^I$ explicitly for your choice of $ I$. What is the characteristic polynomial of $ \Frob _2$ acting on $ E[2]^I$.
    6. What is the characteristic polynomial of $ \Frob _3$ acting on $ E[2]$?

    7. Let $ K$ be a number field. Prove that there is a finite set $ S$ of primes of $ K$ such that

      $\displaystyle \O_{K,S} = \{a \in K^* : \ord _\mathfrak{p}(a\O_K) \geq 0$    all $\displaystyle \mathfrak{p}\not\in S\}
\cup \{0\}
$

      is a prinicipal ideal domain. The condition $ \ord _\mathfrak{p}(a\O_K) \geq 0$ means that in the prime ideal factorization of the fractional ideal $ a\O_K$, we have that  $ \mathfrak{p}$ occurs to a nonnegative power.

    8. Let $ a\in K$ and $ n$ a positive integer. Prove that $ L =
K(a^{1/n})$ is unramified outside the primes that divide $ n$ and the norm of $ a$. This means that if $ \mathfrak{p}$ is a prime of $ \O_K$, and $ \mathfrak{p}$ is coprime to $ n\Norm _{L/K}(a)\O_K$, then the prime factorization of $ \mathfrak{p}\O_L$ involves no primes with exponent bigger than $ 1$.

    9. Write down a proof of Hilbert's Theorem 90, formulated as the statement that for any number field $ K$, we have

      $\displaystyle \H^1(K,\overline{K}^*)=0.
$

  1. Let $ k$ be any field. Prove that the only nontrivial valuations on $ k(t)$ which are trivial on $ k$ are equivalent to the valuation (13.3.3) or (13.3.4) of page [*].
  2. A field with the topology induced by a valuation is a topological field, i.e., the operations sum, product, and reciprocal are continuous.
  3. Give an example of a non-archimedean valuation on a field that is not discrete.
  4. Prove that the field $ \mathbf{Q}_p$ of $ p$-adic numbers is uncountable.
  5. 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].)

  6. In this problem you will compute an example of weak approximation, like I did in the Example 14.3.3. 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$.

  7. 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}$.

  8. Compute the first $ 5$ digits of the $ 10$-adic expansions of the following rational numbers:

    $\displaystyle \frac{13}{2}, \quad \frac{1}{389}, \quad \frac{17}{19},$    the 4 square roots of $41$$\displaystyle .$

  9. Let $ N>1$ be an integer. Prove that the series

    $\displaystyle \sum_{n=1}^{\infty} (-1)^{n+1}n! = 1! - 2! + 3! - 4! + 5! - 6! + \cdots.
$

    converges in $ \mathbf {Q}_N$.

  10. 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}$.

  11. 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.

    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$?

  12. 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)$.]

  13. Find the $ 3$-adic expansion to precision 4 of each root of the following polynomial over $ \mathbf{Q}_3$:

    $\displaystyle f = x^3 - 3x^2 + 2x + 3 \in \mathbf{Q}_3[x].
$

    Your solution should conclude with three expressions of the form

    $\displaystyle a_0 + a_1\cdot 3 + a_2\cdot 3^2 + a_3 \cdot 3^3 + O(3^4).$

    1. Find the normalized Haar measure of the following subset of $ \mathbf{Q}_7^+$:

      $\displaystyle 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^*$.

  14. 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 13 with the weak approximation theorem.]

  15. 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.

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

    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?

  17. 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.

  18. 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$.

  19. 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 nilpotent if there exists $ n>1$ such that $ a^n=0$.)

  20. 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$?

  21. 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.)
  22. Prove Theorem 18.3.5, 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.

William Stein 2012-09-24