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

Euclidean Algorithm in the Ring of Algebraic Integers Generated by the Square Root of Five

This Demonstration shows examples of arithmetic operations in the field , the field of numbers , where and are rational numbers. But instead of using the numbers 1 and , here we use 1 and the golden ratio .

(

5

)a+b

5

a

b

5

ϕ=(1+

1

2

5

)First define the conjugate of to be =x-y. Also define the norm of to be .

z=x+y

5

∈(5

)z

5

z=x+y

5

z=Nx+y-5

z

5

=x+y5

x-y5

=2

x

2

y

An algebraic integer in the field is of the form (a+b, where . If a number is an algebraic integer, its norm is an ordinary integer.

(

5

)1

2

5

)a≡b(mod2)

Write x+y as (x-y)+y(1+, with , , and integers.

1

2

5

1

2

1

2

5

)=a+bϕx

y

a

b

So .

N(a+bϕ)=-

2

a+

b

2

5

4

2

b

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

α

β

(

5

)(

5

)μ=β/α

a+bϕ

a

b

λ=c+dϕ

a-c⩽

1

2

b-d⩽

1

2

N(μ-λ)<1

γ=β-λα

β=λα+γ

N(γ)<N(α)

In the case of division in algebraic integers, we show the pair . This Demonstration shows the Euclidean algorithm in the ring of algebraic integers of :

{λ,γ}

(

5

)β=α+

λ

0

α

1

N()<N(α)

α

1

α=+

λ

1

α

1

α

2

N()<N()

α

2

α

1

α

2

λ

2

α

2

α

3

N()<N()

α

3

α

2

...

α

n-1

λ

n

α

n

α

n+1

N()<N()

α

n+1

α

n

α

n

λ

n+1

α

n+1