Answer to Question #85382 in Abstract Algebra for Narendra Kumar Sharma

Question

Let d∈N , where d ≠1 and d is not divisible by the square of a prime. Prove that
N:Z[ d root(2)]→N∪{0}:N(a+b root(d))=|a^2 −db^2 | satisfies the following properties for x,y∈Z[ droot(2)]:
i) N(x) = 0 ⇔ x = 0
ii) N(xy) = N(x)N(y)
iii) N(x)=1⇔x is a unit
iv) N(x) is prime ⇒ x is irreducible in Z[ d ] .

Answer 1

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