Exercise 24, § 4

An abelian group $G$ is isomorphic to a subgroup of the multiplicative group $K^*$ of nonzero elements of some commutative field $K$ if and only if the torsion subgroup $T$ of $G$ has rank $\leq 1$ (Ex. 22). (For the necessity of this condition, cf. V, p. 79, Prop. 2 ; use the same reference to show that the condition is sufficient for $T$ to be isomorphic to a subgroup of $E^*$, for some field $E$. For all $a \in G/T$ let $g_a$ be a coset representative of $a$ in $G$, then, identifying $T$ with a subgroup of $E^*$, we may write $g_a g_b = e_{\alpha\beta} g_{(\alpha\beta)}$ for all $a, \beta \in G/T$, where $e_{\alpha\beta} \in E$. Show that there exists an $E$-algebra $B$ with basis $(b_i)_{i \in G/T}$ such that $b_a b_b = e_{\alpha\beta} b_{(\alpha\beta)}$ for all $a, \beta$, and prove that $B$ is an integral domain using VI, p. 33, Ex. 20, c). Then the field of fractions $K$ of $B$ satisfies the desired conditions.)