Question #294217

P rove that all real of the form a + b 2 , a; b ∈ Z form s a ring.


1
Expert's answer
2022-02-08T11:20:58-0500

Given, a+b2,a;bZa + b\sqrt2 , 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)

we have:

(a1+b12+a2+b22)+a3+b32=a1+b12+(a2+b22+a3+b32)(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+b=b+aa+b=b+a

we have:

a1+b12+a2+b22=a2+b22+a1+b12a_1+b_1\sqrt2+a_2+b_2\sqrt2=a_2+b_2\sqrt2+a_1+b_1\sqrt2


There is an element 0 in R such that a + 0 = a

we have:

element 0 in R: a=b=0a=b=0

For each a in R there exists −a in R such that a + (−a) = 0

we have:

a+b2+(ab2)=0a+b\sqrt2+(-a-b\sqrt2)=0


2.R is a monoid under multiplication, meaning that:

(ab)c=a(bc)(a ⋅ b) ⋅ c = a ⋅ (b ⋅ c)we have:

((a1+b12)(a2+b22))(a3+b32)=(a1+b12)((a2+b22)(a3+b32))((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))


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=0a=1,b=0


3.Multiplication is distributive with respect to addition, meaning that:

a ⋅ (b + c) = (a ⋅ b) + (a ⋅ c)

we have:

(a1+b12)(a2+b22+a3+b32)=(a1+b12)(a2+b22)+(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)+

+(a1+b12)(a3+b32)+(a_1+b_1\sqrt2)(a_3+b_3\sqrt2)


(b + c) ⋅ a = (b ⋅ a) + (c ⋅ a)

we have:

(a2+b22+a3+b32)(a1+b12)=(a1+b12)(a2+b22)+(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)+

+(a1+b12)(a3+b32)+(a_1+b_1\sqrt2)(a_3+b_3\sqrt2)



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