This paper presents a new construction of the m-fold metaplectic cover
of GLn over an algebraic number field k, where k contains a primitive
m-th root of unity. A 2-cocycle on GLn(A) representing this extension
is given and the splitting of the cocycle on GLn(k) is found explicitly.
The cocycle is smooth at almost all places of k. As a consequence, a
formula for the Kubota symbol on SLn is obtained. The construction
of the paper requires neither class field theory nor algebraic K-theory,
but relies instead on naive techniques from the geometry of numbers
introduced by W. Habicht and T. Kubota. The power reciprocity law
for a number field is obtained as a corollary
