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examples of ring of integers of a number
field∗
alozano†
2013-03-21 18:55:39
Definition 1. Let K be a number field. The ring of integers of K, usually
denoted by OK , is the set of all elements α ∈ K which are roots of some monic
polynomial with coefficients in Z, i.e. those α ∈ K which are integral over Z.
In other words, OK is the integral closure of Z in K.
Example 1. Notice that the only rational numbers which are roots of monic
polynomials with integer coefficients are the integers themselves. Thus, the ring
of integers of Q is Z.
√
Example 2. Let OK denote the ring of integers of K = Q( d), where d is a
square-free integer. Then:
(
√
1+ d
Z
⊕
Z, if d ≡ 1 mod 4,
√2
OK ∼
=
Z ⊕ d Z, if d ≡ 2, 3 mod 4.
In other words, if we let
√
1+ d
,
2
√
(
α=
if d ≡ 1 mod 4,
d, if d ≡ 2, 3 mod 4.
then
OK = {n + mα : n, m ∈ Z}.
Example 3. Let K = Q(ζn ) be a cyclotomic extension of Q, where ζn is a
primitive nth root of unity. Then the ring of integers of K is OK = Z[ζn ], i.e.
OK = {a0 + a1 ζn + a2 ζn2 + . . . + an−1 ζnn−1 : ai ∈ Z}.
Example 4. Let α be an algebraic integer and let K = Q(α). It is not true in
general that OK = Z[α] (as we saw in Example 2, for d ≡ 1 mod 4).
∗ hExamplesOfRingOfIntegersOfANumberFieldi
created: h2013-03-21i by: halozanoi version: h36879i Privacy setting: h1i hExamplei h13B22i
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Example 5. Let p be a prime number and let F = Q(ζp ) be a cyclotomic
extension of Q, where ζp is a primitive pth root of unity. Let F + be the maximal
real subfield of F . It can be shown that:
F + = Q(ζp + ζp−1 ).
Moreover, it can also be shown that the ring of integers of F + is OF + = Z[ζp +
ζp−1 ].
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