Answer to Question #286386 in Abstract Algebra for Arathy

Question #286386

Prove that all real of the form a+b2^1/2,a,b belongs to Z forms a ring

1
Expert's answer
2022-01-24T15:49:53-0500

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)"

we have:

"(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+a"

we have:

"a_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=0"


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

we have:

"a+b\\sqrt2+(-a-b\\sqrt2)=0"


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

"(a \u22c5 b) \u22c5 c = a \u22c5 (b \u22c5 c)"

we have:

"((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=0"


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

a ⋅ (b + c) = (a ⋅ b) + (a ⋅ c)

we have:

"(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\\sqrt2)(a_3+b_3\\sqrt2)"


(b + c) ⋅ a = (b ⋅ a) + (c ⋅ a)

we have:

"(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_1+b_1\\sqrt2)(a_3+b_3\\sqrt2)"


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