Are these infinite groups decomposable? I have been asked to:

Decide whether the following groups are decomposable:
(a) - $(\mathbb{R^*}, \cdot)$
(b) - $(\mathbb{C}, +)$
(c) - $(\mathbb{Q^*}, \cdot)$
(d) - $(\mathbb{Q}, +)$

I would like a hint for item (a). I believe I was able to do itens (b), (c) and (d).
Regarding item (a), I tried to decompose $\mathbb{R^*}$ in rationals and irrationals (but this failed, since the irrationals are not a subgroup) or into algebraic and transcendental numbers (which also fails, since the transcendental numbers are not a subgroup). I also thought about showing that if $\mathbb{R^*} = A \times B$ then $A$ and $B$ do not intersect trivially (thus showing that the group is indecomposable), but I couldn't prove this idea.
Regarding item (b), I decomposed $\mathbb{C}$ into $\mathbb{R}$ and
$i\mathbb{R} = \{iy \ | \ y \in \mathbb{R} \} $.
Regarding item (c), I wrote that $\mathbb{Q^*} = \langle \ p \ | \ p \  \text{is a prime} \rangle = \langle 2 \rangle \ \oplus \ \langle \ p \ | \ p \  \text{is an odd prime} \rangle $.
EDIT: As pointed in the comments, this decomposition is for the multiplicative group of positive rational numbers. A correct decomposition would be, for instance, $\mathbb{Q^*} = \langle 2, -1 \rangle \ \oplus \ \langle \ p \ | \ p \  \text{is an odd prime} \rangle $.
Regarding item (d), I proved that the group is indecomposable by proving that two non-trivial subgroups don't intersect trivially. My reasoning was the same as in: Why is the additive group of rational numbers indecomposable?.
Can anyone give me a hint for item (a)? Thanks in advance.

$(\mathbb R^*, \cdot) = \left(\mathbb R^+, \cdot\right)\times \{1, -1\}$, because (i) $\left(\mathbb R^+, \cdot\right)$ and $\{1, -1\}$ are clearly normal subgroups of $(\mathbb R^*, \cdot)$, (ii) common element of $\left(\mathbb R^+ , \cdot\right)$ and $\{1, -1\}$ is the identity $1$ only, and clearly (iii) $\mathbb R^* = \left(\mathbb R^+\right)\{1, -1\}$. Here $\mathbb R^*$ is the set of all non-zero real numbers and $\mathbb R^+$ is the set of all positive real numbers.
