University of California Ring Isomorphism Injective and Subjective Map HW

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7. Let D be an integer that is not a square. Recall that every element of Q(VD) can be
written as a +bVD for some a, b E Q. Consider the following maps:
o(a + bVD) = a – bVD, Trace(a + bVD) = 2a, Norm(a + bVD) = a? – D62.
(a) (5 points) Show that o is a ring isomorphism. Hint: o oo is the identity map.
(b) (5 points) Prove that the only ring homomorphisms Q(VD) → Q(VD) that send
a → a for all a E Q, are the identity map and o.