Question #162119

Apply the fundamental theorem of 

homomorphism to prove that :

R⁴/R² isomorphic to R²


1
Expert's answer
2021-02-18T13:03:06-0500

ϕ:R4R2\phi: R^4\longrightarrow R^2 is given by (a,b,c,d)(a,b)(a,b,c,d)\mapsto (a,b) This map is trivially homomorphism.

This map is clearly surjective and kernel is given by a=b=0.{a=b=0}. So by homorphism theorem R4/KerϕR2.R^4 /Ker \phi\cong R^2. Now we need to show KerϕR2.Ker \phi\cong R^2. This is given by the map (0,0,x,y)(x,y)(0,0,x,y)\mapsto (x,y) . This map is clearly surjective and its kernel is zero and hence bijective. Homomorphism is trivial.



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