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WOLFRAM|DEMONSTRATIONS PROJECT

Division in the Ring of Algebraic Integers Generated by the Square Root of Five

​
new example
β
α
μ = β/α
λ
δ = β - μα
3+4ϕ
3+2ϕ
7
11
+
6ϕ
11
1+ϕ
-2-3ϕ
This Demonstration illustrates division in the ring of algebraic integers in the field
(
5
)
, that is, the field of numbers
a+b
5
, where
a
and
b
are rational. But instead of using the numbers
1
and
5
, here we use
1
and the golden ratio
ϕ=
1
2
(1+
5
)
.
An algebraic integer in the field
(
5
)
is of the form
1
2
(a+b
5
)
, where
a≡b(mod2)
. Write
1
2
(a+b
5
)
as
1
2
(a-b)+
1
2
b(1+
5
)=c+bϕ
, with
a
,
b
and
c
integers.
The conjugate of a number
a+b
5
is
a-b
5
. The norm is defined by
N(a+b
5
)=(a+b
5
)(a-b
5
)=
2
a
-5
2
b
. So
N(c+dϕ)=
2
c+
d
2
-
5
4
2
d
. If a number is an algebraic integer, its norm is an ordinary integer.
Suppose that
α
and
β
are algebraic integers in
(
5
)
. The quotient
μ=β/α
can be written as
a+bϕ
, where
a
and
b
are rational. Let
λ=c+dϕ
be such that
a-c⩽
1
2
and
b-d⩽
1
2
. Then
N(μ-λ)<1
. If
δ=β-λα
then
β=λα+δ
, where
N(δ)<N(α)
.
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