Download PDF

quadratic imaginary norm-Euclidean
number fields∗
pahio†
2013-03-21 22:24:02
√
Theorem 1. The imaginary quadratic fields Q( d) with d = −1, −2, −3, −7, −11
are norm-Euclidean number fields.
◦
Proof.
d = −2. Any element γ of the
√ 1 . d 6≡ 1 (mod 4), i.e. d = −1 or √
field Q( d) has the canonical
form
γ
=
c
+
c
d where c0 , c1 ∈ Q. We may
0
1
√
write γ = (p + r) + (q + s) d, where p is the rational integer nearest to c0 and
q the one nearest to c1 . So |r| ≤ 12 , |s| ≤ 21 . Thus we may write
√
√
γ = (p + q d) + (r + s d),
| {z } | {z }
δ
κ
where κ is an integer of the field. We then can estimate
√
√
0 ≤ N(δ) = (r+s d)(r−s d) = r2 −s2 d = r2 +s2 |d| ≤
2 2
1
1
3
+2
= < 1,
2
2
4
and√ therefore |N(δ)|
< 1. According to the theorem 1 in the parent entry,
√
Q( −1) and Q( −2) are norm-Euclidean number fields.
2◦ . d ≡ 1 (mod 4), i.e. d ∈ {−3, −7,
−11}. The algebraic integers of
√
√
√
a+b d
Q( d) have now the canonical form
with 2 | a−b. Let γ = c0 + c1 d
2
where c0 , c1 ∈ Q be an arbitrary element of the field. Choose the rational
integer q such that 2q is as close to c1 as possible, i.e. c1 = 2q + s with |s| ≤ 14 ,
and the rational integer t such that 2q + t is as close to c0 as possible; then
p
1
c0 = q+2t
2 + r = 2 + r with |r| ≤ 2 . Then we can write
√
√
√
p
q
p+q d
γ = + r + ( + s) d =
+ (r + s d) .
2
2
2 } | {z }
| {z
κ
δ
∗ hQuadraticImaginaryNormEuclideanNumberFieldsi created: h2013-03-21i by: hpahioi
version: h39126i Privacy setting: h1i hTheoremi h13F07i h11R21i h11R04i
† This text is available under the Creative Commons Attribution/Share-Alike License 3.0.
You can reuse this document or portions thereof only if you do so under terms that are
compatible with the CC-BY-SA license.
1
The number κ is an integer of the field, since p − q = 2t ≡ 0 (mod 2). We get
the estimation
2
2
1
1
15
0 ≤ N(δ) = r2 + s2 |d| ≤
+ 11
=
< 1,
2
4
16
so |N(δ)| < 1. Thus the fields in question are norm-Euclidean number fields.
√Theorem 2. The only quadratic imaginary norm-Euclidean number fields
Q( d) are the ones in which d = −1, −2, −3, −7, −11.
Proof. Let d be any other negative (square-free) rational integer than the
above mentioned ones.
√
√
1◦ . d 6≡ 1 (mod 4). The integers of Q( d) are a + b d where a, b ∈ Z. We
show that there is a number γ that can not be expressed in the form√ γ = κ + δ
with κ an integer of the field and |N(δ)| < 1. Assume that γ := 12 d = κ + δ
√
where κ = a
+ b d is an integer of the field (a, b ∈ Z). Then δ = γ − κ =
√
−a + ( 12 − b) d and N(δ) = |a|2 + |d| · | 12 − b|2 . Because b cannot be 0, we have
| 12 − b| ≥ 12 and thus
|N(δ)| ≥ 0 + |d|
2
1
|d|
5
=
≥ > 1.
2
4
4
√
Therefore Q( d) can not be a norm-Euclidean number field (d = −5, −6, −10
and so on).
√
2◦ . d ≡ 1√ (mod 4). Now |d| ≥ 15. The integers of Q( d) have the
√
form κ = a+b2 d with 2 | a − b. Suppose that γ = 14 + 14 d = κ + δ. Then
√
δ = γ − κ = ( 41 − a2 ) + ( 41 − 2b ) d and
2
2
1 b 2
1 a 2
1
1
|N(δ)| ≥ − + |d| · − ≥
+ 15
= 1.
4 2
4 2
4
4
√
So also these fields Q( d) are not norm-Euclidean number fields.
Remark. The rings of integers of the imaginary quadratic fields of the above
theorems are thus PID’s. There are, in addition, four other imaginary quadratic
fields which are not norm-Euclidean but anyway their rings of integers are PID’s
(see lemma for imaginary quadratic fields, class numbers of imaginary quadratic
fields, unique factorization and ideals in ring of integers, divisor theory).
2