Let Q[ √ 2] = {a + b √ 2 | a, b ∈ Q}
Q[ √ 2] is closed under addition. (a + b √ 2) + (c + d √ 2) = (a + c) + (b + d) √ 2
Q[ √ 2] is closed under multiplication. (a + b √ 2)(c + d √ 2) = ac + ad√ 2 + bc√ 2 + bd√ 2^ 2 = (ac + 2bd) + (ad + bc) √ 2.
• Addition is associative and commutative on Q[ √ 2], since it is associative and commutative in R.
• Identity for addition: 0 = 0 + 0√ 2 ∈ Q[ √ 2].
• Inverses for addition: The inverse of a + b √ 2 is −(a + b √ 2) = −a + −b √ 2 ∈ Q[ √ 2].
• Multiplication is associative and commutative on Q[ √ 2], since it is associative and commutative in R. • Distributivity holds in Q[ √ 2], since it holds in R.
• Identity for multiplication: 1 = 1 + 0√ 2 ∈ Q[ √ 2].
Not only is Q[ √ 2] closed under + and ·, but Q[ √ 2] is a field (a subfield of R).
