Math 538
Assignment 4
Ed Belk
March 12, 2015
1. Let ζ be a primitive p-th root of unity, where p is an odd prime number. Show that Z[ζ]∗ = (ζ)Z[ζ +
ζ −1 ]∗ . Show that
Z[ζ]∗ = {±ζ k (1 + ζ)n : 0 ≤ k < 5, n ∈ Z}
if p = 5. [It is believed that (ζ) denotes the cyclic group of order p generated by ζ. You may assume
without proof the fact that the ring of integers of Q(ζ) is Z[ζ].]
k
2. Let ζ be a primitve m-th root of unity, m ≥ 3. Show that the numbers 1−ζ
1−ζ for (k, m) = 1 are units
in the ring of integers of the field Q(ζ). The subgroup of the group of units they generate is called the
group of cyclotomic units.
3. [rewritten slightly from textbook to lessen ambiguity] If K/L is an extension of number fields and a
and b are ideals of OK , then one has a = aOL ∩ OK , and a|b ⇐⇒ aOL |bOL .
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