Letting $\mathbb{Q}^{\times}$ be the group of non-zero rationals under multiplication, and $\mathbb{Q}$ the additive group of rationals, I am seeking to find the following types of homomorphisms (if, of course, they exist):
a) A surjective homomorphism $f: \mathbb{Q}^{\times} \to \mathbb{Q}$
b) An injective homomorphism $f: \mathbb{Q}^{\times} \to \mathbb{Q}$
I know of a homomorphism $\mathbb{Q}^{\times} \to \mathbb{Z}$ but it is not injective (I refrain from describing it, one can see it here: Non-trivial homomorphism between multiplicative group of rationals and integers). Hints, suggestions, solutions will all be much appreciated.