A Brief Introduction to Classical and Adelic Algebraic ... - William Stein

174 CHAPTER 22. EXERCISES
6. Find the ring of integers of each of the following number fields:
(a) Q( √ −3),
(b) Q( √ 3), and
(c) Q( 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 ↩→ Q be a
field embedding of L into Q that fixes K elementwise. Show that σ extends
in exactly [M : L] ways to a field embedding M ↩→ 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 OK of a number field, and ideals I
and J such that {ab : a ∈ I, b ∈ J} is not an ideal.
11. (a) Let k be a field. Prove that k[x] is a Dedekind domain.
(b) (Problem 1.12 from Swinnerton-Dyer) Let x be an indeterminate. Show
that the ring Z[x] is Noetherian and integrally closed in its field of fractions,
but is not a Dedekind domain.
12. Use Magma to write each of the following (fractional) ideals as a product of
explicitly given prime ideals:
(a) The ideal (2004) in Q( √ −1).
(b) The ideals I = (7) and J = (3) in the ring of integers of Q(ζ7), where ζ7
is a root of the irreducible polynomial x 6 + x 5 + x 4 + x 3 + x 2 + x + 1.
(The field Q(ζ7) is called the 7th cyclotomic field.)
(c) The principal fractional ideal (3/8) in Q( √ 5).
13. Suppose R is an order in the ring OK of integers of a number field. (Recall
that an order is a subring of finite index in OK.) For each of the following
questions, either explain why the answer is yes for any possible order R in any
OK, or find one specific counterexample: