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HW 20. Due November 1. In this assignment, we will see an example of an integral...

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Advance Math

HW 20. Due November 1. In this assignment, we will see anexample of an integral domain that has elements that can befactored as a product of irreducible elements, but thatfactorization is not unique. Let R denote the set of all complexnumbers a + b ? 5i, where a, b ? Z. Let N be the norm on R definedby N(a + b ? 5i) = a 2 + 5b 2 . As before N(z1z2) = N(z1)N(z2), forall z1, z2 ? R. (In fact, this holds for all complex numbers if,for z = c + di ? C we define N(z) = c 2 + d 2 .) (i) Show that R isan integral domain. (ii) Show that the only units in R are ±1.(iii) Use the norm to prove that 2, 3, 1 + ? 5i, 1 ? ? 5i areirreducible elements in R. (iv) Conclude that 6 = 2 · 3 = (1 + ?5i) · (1 ? ? 5i) are two distinct factorizations of 4 into aproduct of irreducible elements.

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