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**HallsofIvy** L= F(u+ v) is the smallest field that contains all members of F and the number u+ v. Any such number can be written as a+ b(u+ v) where a and b are members of F. L(u) is the smallest field that contains all members of L and u. Any such number, then, can be written as (c+ d(u+ v))+ jv where c and d are members of F and j is a member of L, not necessarily a member of F. So any such number is of the form (c+ d(u+ v))+ (e+ f(u+ v))v= c+ d(u+ v)+ ev+ fuv+ fv^2. Of course, uv and v^2 are members of F(u,v) as are c, d(u+v), and ev. So it is clearly true that L(u) is a subset of F(u,v). It remains to be shown that F(u,v) is a subset of L(u).

Any member of F(u,v) is of the form a+ bu+ cv. That is the same as "c+ d(u+ v)+ ev+ fuv+ fv^2= c+ du+ (d+e)v+ fuv+ fv^2" if we take c= a, d= b, e= c- b, and f= 0.