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

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

This Demonstration illustrates division in the ring of algebraic integers in the field , that is, the field of numbers , where and are rational. But instead of using the numbers and , here we use and the golden ratio .

(

5

)a+b

5

a

b

1

5

1

ϕ=(1+

1

2

5

)An algebraic integer in the field is of the form (a+b, where . Write (a+b as (a-b)+b(1+, with , and integers.

(

5

)1

2

5

)a≡b(mod2)

1

2

5

)1

2

1

2

5

)=c+bϕa

b

c

The conjugate of a number is . The norm is defined by . So . If a number is an algebraic integer, its norm is an ordinary integer.

a+b

5

a-b

5

N(a+b-5

5

)=(a+b5

)(a-b5

)=2

a

2

b

N(c+dϕ)=-

2

c+

d

2

5

4

2

d

Suppose that and are algebraic integers in . The quotient can be written as , where and are rational. Let be such that and . Then . If then , where .

α

β

(

5

)μ=β/α

a+bϕ

a

b

λ=c+dϕ

a-c⩽

1

2

b-d⩽

1

2

N(μ-λ)<1

δ=β-λα

β=λα+δ

N(δ)<N(α)