Let I be the ring of integral Hamilton Quaternions and define N: I ->Z by N(a+bi+cj+dk) = a^2 +b^2 +c^2+d^2 a) Prove that N(k)= kk' for all k in I where if k=a+bi+cj+dk then k'=a-bi-cj-dk b)Prove that N(kr)=N(k) N(r) for all k,r in I c) Prove that an element of I is a unit iff it has norm +1

Math Sep 04, 2020

Let I be the ring of integral Hamilton Quaternions and define N: I ->Z by
N(a+bi+cj+dk) = a^2 +b^2 +c^2+d^2
a) Prove that N(k)= kk' for all k in I where if k=a+bi+cj+dk then
k'=a-bi-cj-dk
b)Prove that N(kr)=N(k) N(r) for all k,r in I
c) Prove that an element of I is a unit iff it has norm +1.
Show that I(with multiplication) is isomorphic to the quaternion group of order 8

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