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)

.