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A cyclotomic field (or cyclotomic number field) is a cyclotomic extension of
. These are all of the form
, where is a primitive th root of unity.
The ring of integers of a cyclotomic field always has a power basis over
. Specifically, the ring of integers of
is
.
Given a primitive th root of unity , its minimal polynomial over
is the cyclotomic polynomial . Thus,
, where denotes the Euler phi function.
If is odd, then
. There are many ways to prove this, but the following is a relatively short proof: Since
, we have that
. We also have that
. Thus,
. It follows that
.
Note. If is a positive integer and is an integer such that
, then and
are primitive th roots of unity and generate the same cyclotomic field.
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