# Exercises

1. Let .
1. Find invertible integer matrices and such that is in Smith normal form.
2. What is the group structure of the cokernel of the map defined by multiplication by ?

2. Let be the abelian group generated by with relatoins and . Find a product of cyclic groups that is isomorphic to .

3. Prove that each of the following rings have infinitely many prime ideals:
1. The integers . [Hint: Euclid gave a famous proof of this long ago.]
2. The ring of polynomials over .
3. The ring of polynomials over .
4. The ring 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 denote the subset of all elements of that satisfy a monic polynomial with coefficients in the ring of integers. We proved in class that is a ring.
1. Show that the ideals and in are distinct.
2. Prove that is not Noetherian.

5. Show that neither nor is a unique factorization domain. [Hint: Consider the factorization into irreducible elements of in the first case and  in the second. A nonzero element in a ring is an irreducible element if it is not a unit and if whenever , then one of or is a unit.]

6. Find the ring of integers of each of the following number fields:
1. ,
2. , and
3. .
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 be a finite integral domain. Prove that is a field. [Hint: Show that if is a nonzero element, then has an inverse by considering powers of .]

9. Suppose is a tower of number fields and let be a field embedding of into that fixes elementwise. Show that extends in exactly ways to a field embedding .

1. Suppose and are principal ideals in a ring . Show that the set is an ideal.
2. Give an example of ideals and in the polynomial ring in two variables such that is not an ideal. Your example illustrates why it is necessary to define the product of two ideals to be the ideal generated by .
3. Give an example of a ring of integers of a number field, and ideals  and  such that is not an ideal.

1. Let be a field. Prove that is a Dedekind domain.
2. (Problem 1.12 from Swinnerton-Dyer) Let be an indeterminate. Show that the ring is Noetherian and integrally closed in its field of fractions, but is not a Dedekind domain.

10. Use to write each of the following (fractional) ideals as a product of explicitly given prime ideals:
1. The ideal in .
2. The ideals and in the ring of integers of , where is a root of the irreducible polynomial . (The field is called the th cyclotomic field.)
3. The principal fractional ideal in .

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

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

1. Suppose is a number field of degree . Prove that for some .
2. Prove that if and are two number fields of degree and then .

13. (*) Does there exist a number field  of degree  such that for all ? If so, give an explicit example.

14. Let be the quintic number field generated by a root of . Draw a diagram (be creative) that illustrates the factorization of every prime , with , in .

15. (Problem 1.9 in Swinnerton-Dyer) Show that the only solutions to are given by , as follows. Factor the equation in the number field , which has class number . Show that if is an integer solution then the ideal must be the cube of an ideal, and hence ; thus .

16. Suppose and are ideals in the ring of integers of a number field . Does ? Prove or give a counterexample.

17. Let be the ring of integers , and let

and

be integral ideals of .
1. Find an element of that is congruent to modulo and is congruent to modulo .
2. What is the cardinality of ?
3. Find an element such that is coprime to .

18. Let be the ring of integers of a number field , and suppose has exactly complex embeddings. Prove that the sign of is .

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

20. Find representative ideals for each element of the class group of . Illustrate how to use the Minkowski bound to prove that your list of representatives is complete.

21. Suppose is an order in the ring of integers of a number field. Is every ideal in necessarily generated by two elements?

22. Let be a number field of degree with pairs of complex conjugate embeddings. Prove that

23. Do the exercise on page 19 of Swinnerton-Dyer, which shows that the quantity in the finiteness of class group theorem can be taken to be .

24. Let denote a root of and let . Show that and that has class number (don't just read this off from the output of the commands MaximalOrder and ClassNumber). [Hint: consider the square factors of the discriminant of and show that is an algebra integer if and only if , , and are all even.]

25. If is a closed, bounded, convex, symmetric set in with , for some positive integer , show that contains at least nonzero points in .

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

27. For a given number field , 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).)

1. Prove that there is no number field such that .
2. Is there a number field such that ?

28. Prove that the rank of is unbounded as varies over all number fields.

29. Let .
1. Show that and .
2. Find explicitly generators for the group of units of (you can use for this).
3. Draw an illustration of the log map , including the hyperplane and the lattice in the hyperplane spanned by the image of .

30. Find the group of units of as an abstract group as a function of . (I.e., find the number of cyclic factors and the size of the torsion subgroup. You do not have to find explicit generators!)

31. Let , where is a root .
1. Show that .
2. Find explicitly the log embedding of into a -dimensional hyperplane in , and draw a picture.

32. Prove that if is a quadratic field and the torsion subgroup of has order bigger than , then or .

33. 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.

34. Let and let be a number field. Show that .

35. A totally real number field is a number field in which all embeddings into have image in . Prove there are totally real number fields of degree , for every prime . [Hint: Let denote a primitive th root of unity. For , show that is totally real of degree . Now prove that can be made divisible by any prime.]

36. Give an example of a number field and a prime such that the in the factorization of are not all the same.

37. Let 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 ) in . [The meaning of simplest'' is a matter of taste.]

38. Give examples to show that for a Galois extension, the quantity can be arbirarily large and can be arbitrarily large.

39. Suppose is Galois and is a prime such that is also prime (i.e., is inert in ). Show that is a cyclic group.

40. (Problem 7, page 116, from Marcus Number Fields) For each of the following, find a prime and quadratic extensions and of that illustrates the assertion:
1. The prime can be totally ramified in and without being totally ramified in .
2. The fields and can each contain unique primes lying over while does not.
3. The prime can be inert in and without being inert in .
4. The residue field extensions of can be trivial for and without being trivial for .

41. Let by the symmetric group on three symbols, which has order .
1. Observe that , where is the dihedral group of order , which is the group of symmetries of an equilateral triangle.
2. Use (45a) to write down an explicit embedding .
3. Let be the number field , where is a nontrivial cube root of unity. Show that is a Galois extension with Galois group isomorphic to .
4. We thus obtain a -dimensional irreducible complex Galois representation

Compute a representative matrix of and the characteristic polynomial of for .

42. Let . Show that is Galois over , compute the Galois group of , and compute .

43. Let be any field. Prove that the only nontrivial valuations on which are trivial on are equivalent to the valuation (15.3.3) or (15.3.4) of page .
44. A field with the topology induced by a valuation is a topological field, i.e., the operations sum, product, and reciprocal are continuous.
45. Give an example of a non-archimedean valuation on a field that is not discrete.
46. Prove that the field of -adic numbers is uncountable.
47. Prove that the polynomial has all its roots in , and find the -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].)

48. In this problem you will compute an example of weak approximation, like I did in the Example 16.3.3. Let , let be the -adic absolute value, let be the -adic absolute value, and let be the usual archimedean absolute value. Find an element such that , where , , and .

49. Prove that has a cube root in 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 such that .
2. Suppose . Use induction to show that if and , then there exists such that . (Hint: Show that there is an integer  such that .)
3. Conclude that has a cube root in .

50. Compute the first  digits of the -adic expansions of the following rational numbers:

the 4 square roots of $41$

51. Let be an integer. Prove that the series

converges in .

52. Prove that has a cube root in using the following strategy (this is a special case of Hensel's Lemma'').

1. Show that there is such that .
2. Suppose . Use induction to show that if and , then there exists such that . (Hint: Show that there is an integer  such that .)
3. Conclude that has a cube root in .

53. Let be an integer.
1. Prove that is equipped with a natural ring structure.
2. If is prime, prove that is a field.

1. Let and be distinct primes. Prove that .
2. Is isomorphic to either of or ?

54. Prove that every finite extension of comes from'' an extension of  , in the following sense. Given an irreducible polynomial there exists an irreducible polynomial such that the fields and are isomorphic. [Hint: Choose each coefficient of to be sufficiently close to the corresponding coefficient of , then use Hensel's lemma to show that has a root in .]

55. Find the -adic expansion to precision 4 of each root of the following polynomial over :

Your solution should conclude with three expressions of the form

1. Find the normalized Haar measure of the following subset of :

2. Find the normalized Haar measure of the subset of .

56. Suppose that is a finite extension of and is a finite extension of , with and assume that and have the same degree. Prove that there is a polynomial such that and . [Hint: Combine your solution to 59 with the weak approximation theorem.]

57. Prove that the ring defined in Section 9 really is the tensor product of and , 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.

58. Find a zero divisor pair in .

1. Is a field?
2. Is a field?

59. Suppose denotes a primitive th root of unity. For any prime , consider the tensor product . Find a simple formula for the number of fields appearing in the decomposition of the tensor product . To get full credit on this problem your formula must be correct, but you do not have to prove that it is correct.

60. Suppose and are equivalent norms on a finite-dimensional vector space over a field (with valuation ). Carefully prove that the topology induced by is the same as that induced by .

61. Suppose and are number fields (i.e., finite extensions of ). Is it possible for the tensor product to contain a nilpotent element? (A nonzero element in a ring is if there exists such that .)

62. Let be the number field .

1. In how many ways does the -adic valuation on extend to a valuation on ?
2. Let be a valuation on that extends . Let be the completion of with respect to . What is the residue class field of ?

63. Prove that the product formula holds for similar to the proof we gave in class using Ostrowski's theorem for . You may use the analogue of Ostrowski's theorem for , which you had on a previous homework assignment. (Don't give a measure-theoretic proof.)
64. Prove Theorem 20.3.5, that The global field is discrete in and the quotient of additive groups is compact in the quotient topology.'' in the case when is a finite extension of , where is a finite field.

William Stein 2004-05-06