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number field∗
alozano†
2013-03-21 12:59:40
Definition 1. A field which is a finite extension of Q, the rational numbers, is
called a number field (sometimes called algebraic number field). If the degree
of the extension K/Q is n then we say that K is a number field of degree n
(over Q).
Example 1. The field of rational numbers Q is a number field.
√
integer
Example
2. Let K = Q( d), where d 6= 1 is a square-free non-zero
√
√
2
and
d
stands
for
any
of
the
roots
of
x
−
d
=
0
(note
that
if
d
∈
K
then
√
− d ∈ K as well). Then K is a number field and [K : Q] = 2. We can explictly
describe all elements of K as follows:
√
K = {t + s d : t, s ∈ Q}.
Definition 2. A number field K such that the degree of the extension K/Q is
2 is called a quadratic number field.
In fact, if K is a quadratic number field, then it is easy to show that K is
one of the fields described in Example 2.
Example 3. Let Kn = Q(ζn ) be a cyclotomic extension of Q, where ζn is a
primitive nth root of unity. Then K is a number field and
[K : Q] = ϕ(n)
where ϕ(n) is the Euler phi function. In √
particular, ϕ(3) = 2, therefore K3 is a
quadratic number field (in fact K3 = Q( −3)). We can explicitly describe all
elements of K as follows:
Kn = {q0 + q1 ζn + q2 ζn2 + . . . + qn−1 ζnn−1 : qi ∈ Q}.
In fact, one can do better. Every element of Kn can be uniquely expressed as a
rational combination of the ϕ(n) elements {ζna : gcd(a, n) = 1, 1 ≤ a < n}.
∗ hNumberFieldi
created: h2013-03-21i by: halozanoi version: h31128i Privacy setting:
h1i hDefinitioni h11-00i
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Example 4. Let K be a number field. Then any subfield L with Q ⊆ L ⊆ K
is also a number field. For example, 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 . F + is a number field and it can be shown
that:
F + = Q(ζp + ζp−1 ).
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