Given, a + b 2 , a ; b ∈ Z a + b\sqrt2 , a; b ∈ Z a + b 2 , a ; b ∈ Z
Proof:
A ring is a set R equipped with two binary operations + (addition) and ⋅ (multiplication) satisfying the following three sets of axioms, called the ring axioms
1.R is an abelian group under addition, meaning that:
( a + b ) + c = a + ( b + c ) (a+b)+c=a+(b+c) ( a + b ) + c = a + ( b + c )
we have:
( a 1 + b 1 2 + a 2 + b 2 2 ) + a 3 + b 3 2 = a 1 + b 1 2 + ( a 2 + b 2 2 + a 3 + b 3 2 ) (a_1+b_1\sqrt2+a_2+b_2\sqrt2)+a_3+b_3\sqrt2=a_1+b_1\sqrt2+(a_2+b_2\sqrt2+a_3+b_3\sqrt2) ( a 1 + b 1 2 + a 2 + b 2 2 ) + a 3 + b 3 2 = a 1 + b 1 2 + ( a 2 + b 2 2 + a 3 + b 3 2 )
a + b = b + a a+b=b+a a + b = b + a
we have:
a 1 + b 1 2 + a 2 + b 2 2 = a 2 + b 2 2 + a 1 + b 1 2 a_1+b_1\sqrt2+a_2+b_2\sqrt2=a_2+b_2\sqrt2+a_1+b_1\sqrt2 a 1 + b 1 2 + a 2 + b 2 2 = a 2 + b 2 2 + a 1 + b 1 2
There is an element 0 in R such that a + 0 = a
we have:
element 0 in R: a = b = 0 a=b=0 a = b = 0
For each a in R there exists −a in R such that a + (−a ) = 0
we have:
a + b 2 + ( − a − b 2 ) = 0 a+b\sqrt2+(-a-b\sqrt2)=0 a + b 2 + ( − a − b 2 ) = 0
2.R is a monoid under multiplication, meaning that:
( a ⋅ b ) ⋅ c = a ⋅ ( b ⋅ c ) (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c) ( a ⋅ b ) ⋅ c = a ⋅ ( b ⋅ c ) we have:
( ( a 1 + b 1 2 ) ( a 2 + b 2 2 ) ) ( a 3 + b 3 2 ) = ( a 1 + b 1 2 ) ( ( a 2 + b 2 2 ) ( a 3 + b 3 2 ) ) ((a_1+b_1\sqrt2)(a_2+b_2\sqrt2))(a_3+b_3\sqrt2)=(a_1+b_1\sqrt2)((a_2+b_2\sqrt2)(a_3+b_3\sqrt2)) (( a 1 + b 1 2 ) ( a 2 + b 2 2 )) ( a 3 + b 3 2 ) = ( a 1 + b 1 2 ) (( a 2 + b 2 2 ) ( a 3 + b 3 2 ))
There is an element 1 in R such that a ⋅ 1 = a and 1 ⋅ a = a
we have:
element 1 in R: a = 1 , b = 0 a=1,b=0 a = 1 , b = 0
3.Multiplication is distributive with respect to addition, meaning that:
a ⋅ (b + c ) = (a ⋅ b ) + (a ⋅ c )
we have:
( a 1 + b 1 2 ) ( a 2 + b 2 2 + a 3 + b 3 2 ) = ( a 1 + b 1 2 ) ( a 2 + b 2 2 ) + (a_1+b_1\sqrt2)(a_2+b_2\sqrt2+a_3+b_3\sqrt2)=(a_1+b_1\sqrt2)(a_2+b_2\sqrt2)+ ( a 1 + b 1 2 ) ( a 2 + b 2 2 + a 3 + b 3 2 ) = ( a 1 + b 1 2 ) ( a 2 + b 2 2 ) +
+ ( a 1 + b 1 2 ) ( a 3 + b 3 2 ) +(a_1+b_1\sqrt2)(a_3+b_3\sqrt2) + ( a 1 + b 1 2 ) ( a 3 + b 3 2 )
(b + c ) ⋅ a = (b ⋅ a ) + (c ⋅ a )
we have:
( a 2 + b 2 2 + a 3 + b 3 2 ) ( a 1 + b 1 2 ) = ( a 1 + b 1 2 ) ( a 2 + b 2 2 ) + (a_2+b_2\sqrt2+a_3+b_3\sqrt2)(a_1+b_1\sqrt2)=(a_1+b_1\sqrt2)(a_2+b_2\sqrt2)+ ( a 2 + b 2 2 + a 3 + b 3 2 ) ( a 1 + b 1 2 ) = ( a 1 + b 1 2 ) ( a 2 + b 2 2 ) +
+ ( a 1 + b 1 2 ) ( a 3 + b 3 2 ) +(a_1+b_1\sqrt2)(a_3+b_3\sqrt2) + ( a 1 + b 1 2 ) ( a 3 + b 3 2 )
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