Linear equations with unknowns from a multiplicative group in a function field

Linear Equations with Unknowns from a Multiplicative Group in a Function Field Jan-hendrik Evertse and Umberto Zannier

Let K be a field of characteristic 0, and n an integer > 2. Denote by (K) the n-fold direct product of the multiplicative group K. Thus, the group operation of (K) is coordinatewise multiplication (x1, . . . , xn) · (y1, . . . , yn) = (x1y1, . . . , xnyn). We write (x1, . . . , xn) m := (x1 , . . . , x m n ) for m ∈ Z. We will often denote elements of (K) by bold face characters x, y, etc. Ever...

Linear Equations with Unknowns from a Multiplicative Group Whose Solutions Lie in a Small Number of Subspaces

Abstract. Let K be a field of characteristic 0 and let (K) denote the n-fold cartesian product of K∗, endowed with coordinatewise multiplication. Let Γ be a subgroup of (K) of finite rank. We consider equations (*) a1x1+ · · ·+anxn = 1 in x = (x1, . . . , xn) ∈ Γ, where a = (a1, . . . , an) ∈ (K). Two tuples a,b ∈ (K) are called Γ-equivalent if there is a u ∈ Γ such that b = u · a. Győry and th...

Characterizing the Multiplicative Group of a Real Closed Field in Terms of its Divisible Maximal Subgroup

characterizing the multiplicative group of a real closed field in terms of its divisible maximal subgroup

Properties of Primes and Multiplicative Group of a Field

In the [16] has been proven that the multiplicative group Z/pZ∗ is a cyclic group. Likewise, finite subgroup of the multiplicative group of a field is a cyclic group. However, finite subgroup of the multiplicative group of a field being a cyclic group has not yet been proven. Therefore, it is of importance to prove that finite subgroup of the multiplicative group of a field is a cyclic group. M...