Authors' introduction: In the paper [in: Hahn, Sang Geun (ed.) et al., Recent progress in algebra. Proceedings of an international conference, KAIST, Taejon, South Korea, August 11--15, 1997. Providence, RI: American Mathematical Society. Contemp. Math. 224, 1--27 (1999; Zbl 0939.11035)], \textit{G. W. Anderson} invented a remarkable method of double complex to compute the sign-cohomology of the universal ordinary distribution. \textit{P. Das} [Trans. Am. Math. Soc. 352, No.~8, 3557--3594 (2000; Zbl 1013.11069)] applied Anderson's methods and results to the study of the classical \(\Gamma\)-monomials, and got a series of results greatly illuminating the structure of the Galois group over \(\mathbb Q\) of the extension of \(\mathbb Q(\xi_\infty)\) generated by the algebraic \(\Gamma\)-monomials. Using Anderson's methods, he was also able to give elementary proofs of some facts about algebraic \(\Gamma\)-monomials, which previously could only be proved with the aid of Deligne's theory of absolute Hodge cycles on abelian varieties. In Das' paper, he also proposed some questions for further study. In this paper, we emphasize the applications of distributions in the study of algebraic \(\Gamma\)-monomials. The distribution method introduced by \textit{L. Yin} in [J. Number Theory 80, No.~1, 154--167 (2000; Zbl 1005.11062)] may be applied to any global field. In the largest part of the paper we consider the two cases of the rational number field and of a global function field simultaneously. We explain, among other things, that two criteria for an element to belong to the first or the second sign-cohomology of the universal ordinary distribution are direct consequences of the universality of some distributions constructed from partial zeta functions. This answers \textit{D. Thakur}'s question 9.7(c) in [Ann. Math. (2) 134, No.~1, 25--94 (1991; Zbl 0734.11036)]. In the rational number field case, this gives new proofs of some results in Das' paper, and also gives an affirmative answer to one of the questions listed in the end of Das' paper. In this paper, we also define a new \(\Gamma\)-function in characteristic \(p\). Let \(k\) be a global function field. We fix a place \(\infty\) of \(k\). Let \(\mathbb C_k\) be the completion of the algebraic closure of \(k_\infty\). Thakur defined a characteristic-\(p\) \(\Gamma\)-function from \(\mathbb C_k\) to \(\mathbb C_k\cup\{\infty\}\), and studied this \(\Gamma\)-function carefully. Our new \(\Gamma\)-function is defined on the set of non-zero fractional ideals ofk and takes values in \(\mathbb C_k\cup\{\infty\}\). The advantages of the new \(\Gamma\)-function are that we can consider the Galois action on it and that we can apply the distribution method to it. We study the analogues of the properties for the new \(\Gamma\)-function of the reflection and multiplication formulas of the classical Euler gamma function. As a direct consequence, we get the algebraicity of some monomials of this \(\Gamma\)-function. At the same time, we explain the result of \textit{N. Koblitz} and \textit{A. Ogus} [in: Automorphic forms, representations and L-functions, Proc. Symp. Pure Math. Am. Math. Soc., Corvallis/Oregon 1977, Proc. Symp. Pure Math. 33, Part 2, 343--346 (1979; Zbl 0449.10029)] on the classical \(\Gamma\)-monomials. Finally, we connect algebraic \(\Gamma\)-monomials with the cyclotomic units. In the classical case, this result was first given by Das by using Anderson's double complex. For further properties of this new \(\Gamma\)-function, one needs to develop Anderson's theory on solitons to show the analogue of Deligne's reciprocity, and to develop Anderson-Das' theory on the double complex to study its Galois theoretic characters. The analogue of Deligne reciprocity in the case of the rational function fields has been given by Sinha by using Anderson's solitons. These algebraic \(\Gamma\)-monomials may be a new supply of special units of abelian extensions of a global function field, and may rule out the constant factor in the unit-index formula in [\textit{L. Yin}, Compos. Math. 109, No.~1, 49--66 (1997; Zbl 0902.11023)]. Perhaps this gamma function will also give the answer to Thakur's question (9.7(d) in loc. cit.). In this paper, we also raise two conjectures on the universality of distributions of special values of \(L\)-functions and of our \(\Gamma\)-function. The former is a great generalization of the classical Bass theorem on the cyclotomic units (conjectured by Milnor), and the latter is a characteristic-\(p\) analogue of Rohrlich's conjecture, which is a major unsolved conjecture in transcendental number theory.