Question

已知集合A={a+√2b Ia∈Z,b∈Z,且Ia^2-2b^2I=1},若X∈A,y∈A.求证：（1）XY∈A;(2)

Answer 1

let x= a+√2b

y= a'+√2b'

xy= (a+√2b )(a'+√2b')

= (aa'+2bb')+ (ab'+a'b)√2

let a"= aa'+2bb' ∈Z

b"= ab'+a'b ∈Z

|a"^2-2b"^2|

=|(aa'+2bb')^2 - 2(ab'+a'b)^2|

=| (aa')²+4aa'bb'+4(bb')²- 2(ab')²-4aa'bb'-2(a'b)²|

=| a'²(a²-2b²)-2b'²(a²-2b²) |

=| a'²- 2b'²| ((a²-2b²)=1)

=1 ((a'²- 2b'²)=1)

∴ xy∈ A #

x∈A

x= c+√2d

1/x = 1/(c+√2d)

= (c-√2d)/(c²-2d²)

= c/(c²-2d²) + [ -d/(c²-2d²)]√2

= c-d√2 ( c²-2d² = 1 )

c' = c

d' = -d

|c'²-2d'²|

=|c²-2d²| =1

∴ 1/x∈ A #