Homework for Introduction to Algebraic Number Theory

Unless otherwise stated a ring would mean a commutative ring with identity. If unspecified, $k$ denotes a field and $R$ a ring.

Week 1

1. Let $R$ be a nonzero commutative ring with 1. Let $J \subset I$ be ideals of $R$. Let $\bar I$ denote the ideal $I / J$ of $\bar R=R/J$. Show that $\bar R/\bar I$ is isomorphic to $R/I$. Also show that there is a one to one bijection between ideals of $R$ containing $I$ and ideals of $R/I$.

2. Compute the nilpotents in the ring $\mathbb{Z}/n\mathbb{Z}$ for $n\ge 2$.

3. Let $R$ be an integral domain and $x\in R$. Show that if $x$ is a prime element of $R$ then $x$ is irreducible. Shat the converse is true if $R$ is a UFD and false in general.

4. Let $P_1, P_2, \ldots, P_n$ be prime ideals of a ring $R$. Let $S$ be the complement of $\cup_{i=1}^n P_i$ in $R$. Show that $S$ is a multiplicative subset.

5. Let $S$ be multiplicative subset of a ring $R$. Show that the ring homomorphism $\phi:R\to S^{-1}R$ sending $r$ to the conjugacy class $\frac{r}{1}$ is an isomorphism iff every element of $S$ is a unit in $R$.

Week 2

6. Let $R$ be a ring and $S$ a multiplicative subset. Let $J$ be an ideal of $S^{-1}R$. The subset $\phi^{-1}(J)$ of $R$ is an ideal of $R$ and is (sometimes) denoted by $J\cap R$, where $\phi:R\to S^{1-}R$ is the natural localization map. Show that $J=(J\cap R)S^{-1}R$. Show that this induces a bijection between prime ideals of $S^{-1}R$ and prime ideals of $R$ which do not intersect $S$.

7. Let $q:R\to \bar R$ be surjective ring homomorphism and $S$ be a multiplicative subset of $R$ then $q(S)$ is a multiplicative subset of $\bar R$ and $q(S)^{-1}\bar R$ is isomorphic to $S^{-1}R/\text{ker}(q)S^{-1}R$.

8. Let $R=\mathbb{Z}/120\mathbb{Z}$ and $S=\{ 1, 2, 2^2, 2^3, \ldots\}$ be a multiplicative subset of $R$. Compute $S^{-1}R$. How many elements does it have?

9. Write a proof of Chinese Remainder theorem stated in class.

10. Let $R$ be a ring, $I$ be an ideal and $M$ be an $R$-module. Show that $M/IM$ has a natural $R/I$-module structure. Also show that the natural $R$-module structure on $M/IM$ (viewed as a quotient of an $R$-module by an $R$-submodule) is same as the $R$-module structure obtained via the ring homomorphism $R\to R/I$ and the natural $R/I$-module structure on $M/IM$.

Week 3

11. Let $A\subset B$ be an integral extension. Show that $A[X]\subset B[X]$ is also an integral extension.

12. Let $K$ be a field and $\{R_i\}$ be a collection of integrally closed subrings of $K$. Then show that $R=\cap R_i$ is integrally closed.

13. Let $d$ be a square free integer. Let $R$ be the integral closure of $\mathbb{Z}$ in $\mathbb{Q}(\sqrt{d})$. Show that $R=\mathbb{Z}[\sqrt{d}]$ if $d$ is congruent to 2 or 3 (mod 4) and $R=\mathbb{Z}[\frac{1+\sqrt{d}}{2}]$ otherwise.

14. Let $k$ be a field. Show that the power series ring $k[[x]]$ is a DVR.

15. Let $R$ be a dedikind domain and $P_1,\ldots P_n$ be distinct prime ideals of $R$. Let $y_1,\ldots, y_n\in R$ and $a_1,\ldots,a_n$ be positive integers. Show that there exist $x\in R$ such that $x\equiv y_i (mod P_i^{a_i})$ for all $1\le i \le n$.

Week 4

16. Show that $\mathbb{Z}[\sqrt{-5}]$ is not a UFD. Find an ideal of this ring which is not principal.

17. Let $J\subset I$ be ideals of an integral domain $R$. Show that the following are equivalent:
(i) $J=I$.
(ii) $JR_P=IR_P$ for all prime ideals in $R$.
(iii) $JR_m=IR_m$ for all maximal ideals in $R$.

18. Let $R=\mathbb{Z}[\sqrt{-5}]$. Show that $(41)$ is not a prime ideal. Write it as product of prime ideals.

Week 5

19. Show that every ideal in a DD is generated by two elements.

20. Let $m,n$ be fractional ideals in a DD. Show that $(mn)^{-1}=m^{-1}n^{-1}$.

21. Let $S$ be a multiplicative subset of $R$ and $m,n$ are fractional ideals of $R$.
(i) Show that $S^{-1}m$ is a fractional ideal of $S^{-1}R$.
(ii) Show that $S^{-1}(mn)=(S^{-1}m)(S^{-1}n)$.
(iii) Show that $S^{-1}(m^{-1})=(S^{-1}m)^{-1}$.

22. Let $A$ be a $n\times n$ matrix with entries in a field $K$. Show that the coefficient of $t^{n-1}$ in $det(tI-A)$ is $-trace(A)$.

23. Let $R$ be a normal domain and an integral domain $S$ be an integral extension of $R$. Let $K$ and $L$ be fraction fields of $R$ and $S$ respectively such that $[L:K]<\infty$. Show that $T_{L/K}(S)\subset R$ and $N_{L/K}(S)\subset R$.

Week 6

24. Let $L/K$ be a finite extension and $E\subset L$ be the maximal separable extension of $K$ in $L$. Show that the minimal polynomial of $\alpha \in L \setminus E$ over $E$ is of the form $f(x^p)$ for some polynomial $f(x)\in E[x]$. Use this to show that there exist $q=p^n$ such that for all $x\in L$, $x^q\in E$. $E$ is called the separable closure of $K$ in $L$ and $L/E$ is a purely inseparable extension.

25. Let $R\subset R' \subset R''$ be Dedekind domains and $P''$ be a nonzero prime ideal of $R''$ such that $P=P''\cap R$ is a prime ideal. Then $e(P''/R)=e(P''/R')e(P''\cap R'/R)$ and $f(P''/R)=f(P''/R')f(P''\cap R'/R)$
26. Let $R=\mathbb{Z}[\sqrt{2}]$. Show that for any prime number $p>2$, $e(Q/p)=1$ where $Q$ is a prime ideal of $R$ containing $p$.

27. Let $R$ be an integral domain with fraction field $K$, $M$ be a finitely generated $R$-module and $S=R\setminus 0$. The rank of $M$ is the dimension of $S^{-1}M$ as a $S^{-1}R=K$ vector space.
(a) Let $tM=\{m\in M: rm=0$ for some nonzero $r\in R\}$. Show that $tM$ is a submodule and rank of $M$ is same as rank of $M/tM$.
(b) Show that rank 1 torsion free $R$-module is a fractional ideal of $R$.
(c) When $R$ is a PID and $M$ is torsion free, use induction on rank of $M$ to show that $M$ is free.

Week 7

28. Let $R\subset R'$ be an integral domain such that $R'$ is a finitely generated $R$-module. Show that $R'=\cap\{ S^{-1}R: S=R\setminus P$ where $P$ is a maximal ideal of $R \}$.

29. Find the factorization of the primes 2, 3, 5 in the interal closure of $\mathbb{Z}$ in $\mathbb{Q}(\sqrt[3]{2})$.

30. Let $K=\mathbb{Q}(\theta)$ of dimension $n$ over $\mathbb{Q}$, $R'$ the integral closure of $\mathbb{Z}$ in $K$ and assume $\theta\in R'$. Show that $R'=\mathbb{Z}[\theta]$ iff $\Delta(\theta)=\Delta(R/Z)$. Use it to compute $\Delta(R'/\mathbb{Z})$ where $R'$ is the integral closure of $\mathbb{Z}$ in $\mathbb{Q}(\sqrt{d})$ where $d$ is a square free integer.

Week 8

31. Let $m_1,\ldots, m_r$ be pairwise relatively prime, square-free integerswith all or all but one congruent to 1 mod 4. Determine the ring of integers in $\mathbb{Q}(\sqrt{m_1},\ldots,\sqrt{m_r})$.

32. Show that the ring of integers of $\mathbb{Q}(\sqrt{6},\sqrt{14})$ is strictly bigger that $\mathbb{Z}[\sqrt{6},\sqrt{14}]$.

33. Let $L$ and $M$ be two finite field extensions of $K$. Assume both subsets of the algebraic closure $\bar K$ are linearly disjoint over $K$. Let $\tilde L$ and $\tilde M$ be the Galois closure of $L/K$ and $M/K$. Show that $\tilde L$ and $\tilde M$ are linearly disjoint over $K$.

34. Janusz, page 57 Exercise 2.

35. Janusz, page 58 Exercise 7.

Week 8

36. Janusz, page 58 Exercise 3,4,5,6.

Week 9

37. Janusz, page 62 Exercise 1, 3, 4.

38. Janusz, page 78 Exercise 2; page 81 Exercise 3, 4.