# Exercises

1. Let .
1. Find the Smith normal form of .
2. Prove that the cokernel of the map given by multiplication by  is isomorphic to .

2. Show that the minimal polynomial of an algebraic number is unique.

3. Which of the following rings have infinitely many prime ideals?
1. The integers .
2. The ring of polynomials over .
3. The quotient ring .
4. The ring of polynomials over the ring .
5. The quotient ring , for a fixed positive integer .
6. The rational numbers  .
7. The polynomial ring in three variables.

4. Which of the following numbers are algebraic integers?
1. The number .
2. The number .
3. The value of the infinite sum .
4. The number , where is a root of .

5. Prove that is not noetherian.

6. Let .
1. Is an algebraic integer?
2. Explicitly write down the minimal polynomial of as an element of .

7. Which are the following rings are orders in the given number field.
1. The ring in the number field .
2. The ring in the number field .
3. The ring in the number field .
4. The ring in the number field .

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

9. Let be a field.
1. Prove that the polynomial ring is a Dedekind domain.
2. Is a Dedekind domain?

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

1. Give an example of two ideals in a commutative ring whose product is not equal to the set .
2. Suppose is a principal ideal domain. Is it always the case that

for all ideals in ?

11. Is the set of rational numbers with denominator a power of 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 be the ring of integers of a number field. Let  denote the abelian group of fractional ideals of .
1. Prove that is torsion free.
2. Prove that is not finitely generated.
3. Prove that is countable.
4. Conclude that if and are number fields, then there exists some (non-canonical) isomorphism of groups .

14. From basic definitions, find the rings of integers of the fields and .

15. In this problem, you will give an example to illustrate the failure of unique factorization in the ring of integers of .
1. Give an element that factors in two distinct ways into irreducible elements.
2. Observe explicitly that the 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 into prime ideals.

16. Factor the ideal as a product of primes in the ring of integers of . You're allowed to use a computer, as long as you show the commands you use.

17. Let be the ring of integers of a number field , and let be a prime number. What is the cardinality of in terms of and , where is the ideal of generated by ?

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 .
4. An order in the ring of integers of a number field of degree .
5. The matrix on of left multiplication by an element of , where  is a degree  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 be a homomorphism of (commutative) rings.
1. Prove that if is an ideal, then is an ideal of .
2. Prove moreover that if is prime, then is also prime.

20. Let be the ring of integers of a number field. The Zariski topology on the set of all prime ideals of has closed sets the sets of the form

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

21. Explicitly factor the ideals generated by each of , , and in the ring of integers of . (Thus you'll factor separate ideals as products of prime ideals.) You may assume that the ring of integers of is , but do not simply use a computer command to do the factorizations.

22. Let ,where is a primitive th root of unity. Note that  has ring of integers .
1. Factor , , , , , and in the ring of integers . You may use a computer.
2. For , find a conjectural relationship between the number of prime ideal factors of and the order of the reduction of  in .
3. Compute the minimal polynomial of . Reinterpret your conjecture as a conjecture that relates the degrees of the irreducible factors of to the order of modulo . 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: , .
2. Find the ring of integers of , where using a computer.

23. Let be a prime. Let be the ring of integers of a number field , and suppose is such that is finite and coprime to . Let be the minimal polynomial of . We proved in class that if the reduction of factors as

where the are distinct irreducible polynomials in , then the primes appearing in the factorization of are the ideals . In class, we did not prove that the exponents of these primes in the factorization of are the . Prove this.

24. Let , , and as elements of .
1. Prove that the ideals , , and are coprime in pairs.
2. Compute .
3. Find a single element in that is congruent to  modulo , for each .

25. Find an example of a field of degree at least  such that the ring of integers of is not of the form for any .

26. Let be a prime ideal of , and suppose that is a finite field of characteristic . Prove that there is an element such that . This justifies why we can represent prime ideals of as pairs , as is done in SAGE. (More generally, if is an ideal of , we can choose one of the elements of to be any nonzero element of .)

27. (*) Give an example of an order in the ring of integers of a number field and an ideal such that  cannot be generated by elements as an ideal. Does the Chinese Remainder Theorem hold in ? [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 contained in its ring of integers:

and

any extension of of degree . [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 field must be large.]

29. Prove that the quantity in our theorem about finiteness of the class group can be taken to be , as follows (adapted from [SD01, pg. 19]): Let be the set of elements such that

1. Prove that is convex and that , where

[Hint: For convexity, use the triangle inequality and that for , we have

for . In polar coordinates this last inequality is

which is trivial. That follows from the inequality between the arithmetic and geometric means.
2. Transforming pairs from Cartesian to polar coordinates, show also that , where

and is given by ( ), ( ) and

3. Prove that

and deduce by induction that

30. Let  vary through all number fields. What torsion subgroups actually occur?

31. If , we say that has rank . Let  vary through all number fields. What ranks actually occur?

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

33. Let .
1. Show that and .
2. Find explicit generators for the group of units .
3. Draw an illustration of the log map , including the hyperplane and the lattice in the hyperplane spanned by the image of .

34. Let be a number field. Prove that if and only if  ramifies in . (Note: This fact is proved in many books.)

35. Using Zorn's lemma, show that there are homomorphisms with finite image that are not continuous, since they do not factor through the Galois group of any finite Galois extension. [Hint: The extension is an extension of  with Galois group . The index-two open subgroups of  correspond to the quadratic extensions of  . However, Zorn's lemma implies that  contains many index-two subgroups that do not correspond to quadratic extensions of  .]

1. Give an example of a finite nontrivial Galois extension of and a prime ideal such that .
2. Give an example of a finite nontrivial Galois extension of and a prime ideal such that has order .
3. Give an example of a finite Galois extension  of and a prime ideal such that is not a normal subgroup of .
4. Give an example of a finite Galois extension  of and a prime ideal such that is not a normal subgroup of .

36. 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 (39a) 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 .

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

is an isomorphism.

38. Suppose is a finite group and is a finite -module. Prove that for any , the group is a torsion abelian group of exponent dividing the order of .

39. Let and let be the group of units of , which is a module over the group . Compute the cohomology groups and . (You shouldn't use a computer, except maybe to determine .)

40. Let and let  be the class group of , which is a module over the Galois group . Determine and .

41. Let be the elliptic curve . Let be the group of points of order dividing  on . Let

be the mod  Galois representation associated to .
1. Find the fixed field of .
2. Is surjective?
3. Find the group .
4. Which primes are ramified in ?
5. Let be an inertia group above , which is one of the ramified primes. Determine explicitly for your choice of . What is the characteristic polynomial of acting on .
6. What is the characteristic polynomial of acting on ?

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

all

is a prinicipal ideal domain. The condition means that in the prime ideal factorization of the fractional ideal , we have that  occurs to a nonnegative power.

8. Let and a positive integer. Prove that is unramified outside the primes that divide  and the norm of . This means that if is a prime of , and is coprime to , then the prime factorization of involves no primes with exponent bigger than .

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

1. Let be any field. Prove that the only nontrivial valuations on which are trivial on 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 of -adic numbers is uncountable.
5. 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].)

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

7. 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 .

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

the 4 square roots of $41$

9. Let be an integer. Prove that the series

converges in .

10. 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 .

11. 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 ?

12. 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 .]

13. 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 .

14. 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 13 with the weak approximation theorem.]

15. 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.

16. Find a zero divisor pair in .

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

17. 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.

18. 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 .

19. 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 nilpotent if there exists such that .)

20. 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 ?

21. 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.)
22. Prove Theorem 18.3.5, 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 2012-09-24