2. Extending the Rationals with the Square Root of Five

add

subtract

multiply

divide

divide in integers

show result

new example

1+2ϕ

2+2ϕ

3+4ϕ

This Demonstration shows examples of arithmetic operations in the extended field

(

5

)

, that is, in the field of numbers

p+q

5

, where

p

and

q

are rational numbers. But instead of using the numbers

1

and

5

, here we use

1

and the golden ratio

ϕ=

1

2

(1+

5

)

.

First define the conjugate of

z=x+y

5

∈

to be

z

=x-y

5

. Also define the norm of

z=x+y

5

to be

z

=Nx+y

5

=x+y

5

x-y

5

=

2

x

-5

2

y

.

An algebraic integer in the field

(

5

)

is of the form

1

2

(a+b

5

)

, where

a≡b(mod2)

. If a number is an algebraic integer, its norm is an ordinary integer.

Write

1

2

x+y

5



as

1

2

(x-y)+

1

2

y(1+

5

)=a+bϕ

, with

x

,

y

,

a

and

b

integers.

So

N(a+bϕ)=

2

a+

b

2

-

5

4

2

b

.

Suppose that

α

and

β

are algebraic integers in

(

5

)

. In the field

(

5

)

, the quotient

μ=β/α

can be written as

a+bϕ

, where

a

and

b

are rational. Now we want to get the quotient in the ring of algebraic integers. The procedure is this: Let

λ=c+dϕ

, where

c

and

d

are ordinary integers such that

a-c⩽

1

2

and

b-d⩽

1

2

. Then

N(μ-λ)<1

. The reminder is

γ=β-λα

. So we write

β=λα+γ

, where

N(γ)<N(α)

. The button "division in integers" shows the pair

{λ,γ}

, the quotient and reminder.