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}