Math 538
Assignment 4
Due Thursday, March 12, 2014
1. Show that in every ideal a 6= 0 of OK there exists an a 6= 0 such that |NK/Q (a)| ≤ M [OK : a], where
s p
|dK | is the so-called Minkowski bound.
M = nn!n π4
2. How many integral ideals a / Z[i] are there with a given norm, N (a) = n? Your answer will depend on the
prime factors of n.
3. Show that the absolute value of the discriminant |dK | tends to ∞ with the degree n of the field extension.
√
4. Let a be an integral ideal of K and am = (a). Show that a becomes a principal ideal in the field L = K( m a),
in the sense that aOL = (α).
5. Show that, for every number field K, there exists a finite extension such that every ideal of K becomes a
principal ideal.
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