Question

4 Explain intuitively why Z[V2] Z[V3]. Back your intuition with a proof. [Note: this example not only says that a +bv2 a +bv3 is not an isomorph ism. It says that no isomorphism can be found at all-no matter how clever a choice of mapping you might try to make.] Hint; Intuition: ZV2] has an element whose square is 1+1 (i.e. 2); Z[3] surely hasn't? Proof: For any isomorphism 0 we'd have 0(1)-1 hence (2)=2. Suppose 02)=a+ b/3. Then 2=(a+b/3)

Answer 1

Q4 Aus We conhradichion Tposeille let, ZiG] ZIB), henca, thee exigt omorphim :ZL]Z[B], an Sinu 9tis om Aso, 2Nfi So, 2. (2.) (2) Again 1+! = 2 . OO +9(1) (2) 2 From omd i wequto) 2. Sinu = o+1of t Z[£J We eom assum ニ a+3 a, be 2+alfs 3 -+312l 3 Form ii omd we have / C3 2ab 3 2 = thus も30 a= omd 土, Henu .mhatichn buk