Anderson's double complex and gamma monomials for rational function fields

We investigate algebraic $\Gamma$-monomials of Thakur's positive characteristic $\Gamma$-function, by using Anderson and Das' double complex method of computing the sign cohomology of the universal ordinary distribution. We prove that the $\Gamma$-monomial associated to an element of the second sign cohomology of the universal ordinary distribution of $\mathbb{F} _{q}(T)$generates a Kummer extension of some Carlitz cyclotomic function field, which is also a Galois extension of the base field $\mathbb{F} _{q}(T)$. These results are characteristic-$p$ analogues of those of Deligne on classical $\Gamma$-monomials, proofs of which were given by Das using the double complex method. In this paper, we also obtain some results on $e$-monomials of Carlitz's exponential function.