Final answer:
To show that the norm function N: Z[√(-5)] -> Z is multiplicative, we need to prove that for any two elements a and b in Z[√(-5)], their product N(ab) is equal to the product of their norms N(a) and N(b). To prove this, we expand the product ab and calculate its norm, while also calculating the norms of a and b. The proof shows that N(ab) = N(a)N(b), establishing the multiplicative property of the norm function.
Step-by-step explanation:
To show that the norm function N: Z[√(-5)] -> Z is multiplicative, we need to prove that for any two elements a = m + n√(-5) and b = p + q√(-5) in Z[√(-5)], their product N(ab) is equal to the product of their norms N(a) and N(b).
Let's expand the product ab and calculate its norm:
If a = m + n√(-5) and b = p + q√(-5), then ab = (m + n√(-5))(p + q√(-5)).
Expanding this product, we get:
ab = mp + mq√(-5) + np√(-5) + 5nq.
Simplifying, we have:
ab = (mp + 5nq) + (mq + np)√(-5).
The norm of ab, N(ab), is defined as the sum of the squares of the coefficients of the terms without the square root of -5:
N(ab) = (mp + 5nq)2 + (mq + np)2.
Similarly, the norms of a, N(a), and b, N(b), are:
N(a) = m2 + 5n2 and N(b) = p2 + 5q2.
To prove that N is multiplicative, we need to show that N(ab) = N(a)N(b):
N(ab) = (mp + 5nq)2 + (mq + np)2 = (m2 + 5n2)(p2 + 5q2) = N(a)N(b).