# 2. Extending the Rationals with the Square Root of Five

2. Extending the Rationals with the Square Root of Five

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

(

5

)p+q

5

p

q

1

5

1

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

∈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. Now we want to get the quotient in the ring of algebraic integers. The procedure is this: Let , where and are ordinary integers such that and . Then . The reminder is . So we write , where . The button "division in integers" shows the pair , the quotient and reminder.

α

β

(

5

)(

5

)μ=β/α

a+bϕ

a

b

λ=c+dϕ

c

d

a-c⩽

1

2

b-d⩽

1

2

N(μ-λ)<1

γ=β-λα

β=λα+γ

N(γ)<N(α)

{λ,γ}