Math 627
Homework #3
October 13, 2014
Due Wednesday, October 22
1. Suppose that A is a Dedekind domain that is also a local ring (i.e., it has a unique
maximal ideal). Show that A is a principal ideal domain. The following steps may be
useful: let m be the maximal ideal of A,
(a) Show there exists some element π ∈ m \ m2 ;
(b) Show that m = (π);
(c) Show that every ideal of A is of the form (π n ), n ≥ 0.
2. Let K be a number field with [K : Q] = n, let M ⊆ OK be a Z-submodule of rank n,
and let α1 , . . . , αn ∈ M be a Z-basis of M . Show that if p is a prime number such that
p | [OK : M ], then p2 | DQK (α1 , . . . , αn ).
3. Let K be a number field with [K : Q] = n, and suppose K = Q(α) with α ∈ OK .
Let p be a prime number, and suppose that the irreducible polynomial of α over Q is
an Eisenstein polynomial with respect to p. Show that p does not divide |OK /Z[α]|.
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4. Use problems 2 & 3 to prove that if K = Q( 3 2), then OK = Z[ 3 2].
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5. (a) Show that Q( −2) and Q( 7) have trivial class groups.
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(b) Show that Q( 3 2) has trivial class group.
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(c) Show that the class number of Q( −6) is 2.
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