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norm-Euclidean number field∗
pahio†
2013-03-21 22:23:48
Definition. An algebraic number field K is a norm-Euclidean number field,
if for every pair (α, β) of the integers of K, where β 6= 0, there exist integers
κ and % of the field such that
|N(%)| < |N(β)|.
α = κβ + %,
Here N means the norm function in K.
Theorem 1. A field K is norm-Euclidean if and only if each number γ of
K is expressible in the form
γ = κ+δ
(1)
where κ is an integer of the field and |N(δ)| < 1.
Proof. First assume the condition (1). Let α and β be integers of K, β 6= 0.
Then there are the numbers κ, δ ∈ K such that κ is integer and
α
= κ + δ,
β
|N(δ)| < 1.
Thus we have
α = κβ + βδ = κβ + %.
Here % = βδ is integer, since α and κβ are integers. We also have
|N(%)| = |N(β)| · |N(δ)| < |N(β)| · 1 = |N(β)|.
Accordingly, K is a norm-Euclidean number field. Secondly assume that K is
norm-Euclidean. Let γ be an arbitrary element of the field. We can determine
a rational integer m (6= 0) such that mγ is an algebraic integer of K. The
assumption guarantees the integers κ, % of K such that
mγ = κm + %,
N(%) < N(m).
∗ hNormEuclideanNumberFieldi
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Thus
γ=
mγ
%
=κ+ ,
m
m
% |N(%)|
< 1,
=
N
m
|N(m)|
Q.E.D.
Theorem 2. In a norm-Euclidean number field, any two non-zero integers
have a greatest common divisor.
Proof. We recall that the greatest common divisor of two elements of a commutative ring means such a common divisor of the elements that it is divisible
by each common divisor of the elements. Let now %0 and %1 be two algebraic
integers of a norm-Euclidean number field K. According the definition there
are the integers κi and %i of K such that

%0 = κ2 %1 + %2 , |N(%2 )| < |N(%1 )|





%1 = κ3 %2 + %3 , |N(%3 )| < |N(%2 )|



% = κ % + % , |N(% )| < |N(% )|
2
4 3
4
4
3

·
·
·
·
·
·




%n−2 = κn %n−1 + %n , |N(%n )| < |N(%n−1 )|



%n−1 = κn+1 %n + 0,
The chain ends to the remainder 0, because the numbers |N(%i )| form a descending sequence of non-negative rational integers — see the entry norm and trace
of algebraic number. As in the Euclid’s algorithm in Z, one sees that the last
divisor %n is one greatest common divisor of %0 and %1 . N.B. that %0 and %1
may have an infinite amount of their greatest common divisors, depending the
amount of the units in K.
Remark. The ring of integers of any norm-Euclidean number field is a
unique factorization domain and thus all ideals of the ring are principal ideals.
But not all algebraic
√ number fields with ring of integers a UFD are normEuclidean, e.g. Q( 14).
√
Theorem 3. The only norm-Euclidean quadratic fields Q( d) are those
with
d ∈ {−11, −7, −3, −2, −1, 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73}.
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