Hi, could you please help with the following question. I need to understand the method in proving it's a homomorphism and then i can do other similar questions i've been asked to complete.
Let Z x Z be the ring which is the cartesian product of two copies of Z. Determine if the following is a ring homomorphism.
ZxZ -> Z, h(x,y)=x
(Z is the set of integers, and -> is an arrow)
Thanks for you help.
Thanks so much for your help.
I have another question on rings and fields;
Let F be the set of all complex numbers of the form a+bi with a, b in Q. (Q= rationals) Show that F is a subring of C (C=complex numbers), and that it is a field. Show that F is isomprphic to the field of fractions of the ring Z[i] of Gaussian integers.
I have shown that F is a subring of the complex numbers using the subring test.
Is it correct to prove F is a field by showing it has a one (1), is not the zero ring, every non-zero number has an inverse (a/(a^2 +b^2) -(b/(a^2 +b^2))i) and that it is commutative (a+bi) +(c+di) = (c+di) + (a+bi)?
Finally is showing that F is isomorphic to the field of fractions of the ring Z[i] of Gaussian integers the same as showing that the Guassian numbers are isomorphic to the field of Gaussian rationals.
How do i show that these are isomorphic?
Thanks a lot for your help.