Prove that every field of characteristic 0 is infinite. [Hint: Consider the elements n1F with n in Z, n>0] Solution Consider a map from Z to F given by f(n)=n.1F .Here F is a field of charecteristic zero Claim: f is an isomorphism of Z into F.that is to prove that f(n1+n2)=f(n1)+f(n2) f(n1n2)=f(n1)f(n2) (i)f(n1+n2)=(n1+n2)1F =(n11F+n21F) =f(n1)+f(n2) (ii)f(n1n2)=(n1n2)1F =(n1n2)12F =(n11F)(n21F) =f(n1)+f(n2) Therefore f is homomorphism Also if f(n)=0 then n.1F=0 As F is of charecteristic zero and 1F is non zero n=0 therefore kernel f =0 This implies that the function is a one to one function. Therefore there exists an isomorphism from Z into F and Z is infinite which implies that F must also be infinite as range of f lies inside F.