  The problem of computing \emph{the exponent lattice} which consists of all
the multiplicative relations between the roots of a univariate polynomial has
drawn much attention in the field of computer algebra. As is known, almost all
irreducible polynomials with integer coefficients have only trivial exponent
lattices. However, the algorithms in the literature have difficulty in proving
such triviality for a generic polynomial. In this paper, the relations between
the Galois group (respectively, \emph{the Galois-like groups}) and the
triviality of the exponent lattice of a polynomial are investigated. The
$\bbbq$\emph{-trivial} pairs, which are at the heart of the relations between
the Galois group and the triviality of the exponent lattice of a polynomial,
are characterized. An effective algorithm is developed to recognize these
pairs. Based on this, a new algorithm is designed to prove the triviality of
the exponent lattice of a generic irreducible polynomial, which considerably
improves a state-of-the-art algorithm of the same type when the polynomial
degree becomes larger. In addition, the concept of the Galois-like groups of a
polynomial is introduced. Some properties of the Galois-like groups are proved
and, more importantly, a sufficient and necessary condition is given for a
polynomial (which is not necessarily irreducible) to have trivial exponent
lattice.
