Math 129: Algebraic Number Theory Homework Assignment 1

William Stein

Due: Thursday, February 19, 2004

Notes:

• Unless otherwise noted, if you can figure out how to use a computer program to solve a problem, please do. For complete credit you must describe exactly how you used the computer (commands typed, output, etc.) You might find

 http://modular.fas.harvard.edu/calc/ useful.

• You are allowed to work with other people on homework problems, but you must acknowledge their assistance.

• Copying a homework solution if you find it in a book is allowed, but you must reword it in your own way and cite your sources. Learning to use the literature is valuable.

• If you have questions, email me at was@math.harvard.edu.

The problems:

1. Let A = (\begin{matrix}4 & 7 & 2 \\ 2 & 4 & 6 \\ 0 & 0 & 0 \end{matrix}).

(a) Find invertible integer matrices P and Q such that P AQ is in Smith normal form.

(b) What is the group structure of the cokernel of the map Z^{3} → 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:

(a) The integers Z. [Hint: Euclid gave a famous proof of this long ago.]

(b) The ring Q[x] of polynomials over Q.

(c) The ring Z[x] of polynomials over Z.

(d) The ring \overline{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{Z} denote the subset of all elements of \overline{Q} that satisfy a monic polynomial with coefficients in the ring Z of integers. We proved in class that \overline{Z} is a ring.

(a) Show that the ideals (2) and (\sqrt{2}) in \overline{Z} are distinct.

(b) Prove that \overline{Z} is not Noetherian.

5. Show that neither Z[\sqrt{−6}] nor 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:

(a) Q(\sqrt{−3}),

(b) Q(\sqrt{3}), and

(c) 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 ⊂ L ⊂ M is a tower of number fields and let σ : L ↪ \overline{Q} be a field embedding of L into \overline{Q} that fixes K elementwise. Show that σ extends in exactly [M : L] ways to a field embedding M ↪ \overline{Q}.

10. (a) Suppose I and J are principal ideals in a ring R. Show that the set {ab : a ∈ I, b ∈ J} is an ideal.

(b) Give an example of ideals I and J in the polynomial ring Q[x, y] in two variables such that {ab : a ∈ I, b ∈ 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 ∈ I, b ∈ J}.

(c) Give an example of a ring of integers O_{K} of a number field, and ideals I and J such that {ab : a ∈ I, b ∈ J} is not an ideal.