Originally Posted by **rgep**

Since each of the fields I mentioned has the property that the corresponding ring over the integers (its maximal order or ring of integers) has unique factorisation, every element of the field is a unit (and here the only possibilities are +- 1) times a product of prime elements. Since in fact the same is true of the rationals you may prefer to think of that as your example. So every element of the multiplicative group is uniquely expressible as $\displaystyle (-1)^{e_0} \cdot \pi_1^{e_1} \cdot \pi_2^{e_2} \cdots$, where the $\displaystyle \pi_i$ are the prime elements, $\displaystyle e_0$ is either 0 or 1 (taken mod 2) and the $\displaystyle e_i$ are integers, with the property that only finitely many of them are non-zero.