Answer to Question #133618 in Abstract Algebra for PRATHIBHA ROSE C S

Question #133618
Prove that Z[√2]={a+b√2;a,b element of Z} is a ring under the ordinary addition and multiplication
1
Expert's answer
2020-09-27T16:38:32-0400

"\\mathbb{Z}[\\sqrt{2}]" is a ring under the ordinary addition and multiplication because:

1) "\\mathbb{Z}[\\sqrt{2}]" is closed under addition

If "x,y\\in \\mathbb{Z}[\\sqrt{2}]" , then "x=a+b\\sqrt{2}" and "y=c+d\\sqrt{2}" , where "a,b,c,d\\in \\mathbb{Z}" .

"x+y=(a+b\\sqrt{2})+(c+d\\sqrt{2})=(a+c)+(b+d)\\sqrt{2}\\in \\mathbb{Z}[\\sqrt{2}]"

2) addition is associative ( "\\mathbb{Z}[\\sqrt{2}]" is a subset of "\\mathbb{R}" )

3) there exists "0=0+0\\cdot \\sqrt{2}\\in \\mathbb{Z}[\\sqrt{2}]"

4) "\\forall x\\in \\mathbb{Z}[\\sqrt{2}],\\ x=a+b\\sqrt{2}, \\ \\ \\exists (-x)\\in \\mathbb{Z}[\\sqrt{2}], \\ -x=(-a)+(-b)\\sqrt{2}"

5) addition is commutative ("\\mathbb{Z}[\\sqrt{2}]" is a subset of "\\mathbb{R}" )

6) "\\mathbb{Z}[\\sqrt{2}]" is closed under multiplication

If "x,y\\in \\mathbb{Z}[\\sqrt{2}]" , then "x=a+b\\sqrt{2}" and "y=c+d\\sqrt{2}" , where "a,b,c,d\\in \\mathbb{Z}" .

"x\\cdot y=(a+b\\sqrt{2})(c+d\\sqrt{2})=(ac+2bd)+(ad+bc)\\sqrt{2}\\in \\mathbb{Z}[\\sqrt{2}]"

7) multiplication is associative ("\\mathbb{Z}[\\sqrt{2}]" is a subset of "\\mathbb{R}" )

8) distributivity ("\\mathbb{Z}[\\sqrt{2}]" is a subset of "\\mathbb{R}" )



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