Let $F$ be any field of zero characteristic, $F^{\ast}$ its multiplicative group and $T$ is the torsion group. 
Is it true that $T$ is a direct summand for $F^{\ast}$?



torsion group of the multiplicative group of a field
This was a problem that was asked by Fuchs in his book "Abelian groups" (1958). It was first solved in negative by P. M. Cohn in "Eine Bemerkung uber die multiplikative Gruppe eines Korpers", Arch. Math. (Basel) 13 (1962) 344-48. ([MR0146252](http://www.ams.org/mathscinet-getitem?mr=146252)). Later W. May gave a counterexample as an algebraic extension of $\mathbb Q$, in "Multiplicative groups of fields", Proc. London Math. Soc. (3) 24 (1972), 295–306. ([MR0294490](http://www.ams.org/mathscinet-getitem?mr=294490))


