multiplicatively independent

A set X of nonzero complex numbersMathworldPlanetmathPlanetmath is said to be multiplicatively independent iff every equation

x1ν1x2ν2xnνn= 1

with  x1,x2,,xnX  and  ν1,ν2,,νn  implies that

ν1=ν2==νn= 0.

For example, the set of prime numbersMathworldPlanetmath is multiplicatively independent, by the fundamental theorem of arithmeticsMathworldPlanetmath.

Any algebraically independentMathworldPlanetmath set is also multiplicatively independent.

Evidently, {x1,x2,,xn} is multiplicatively independent if and only if the numbers logx1, logx2, …, logxn are linearly independentMathworldPlanetmath over .  Thus the Schanuel’s conjecture may be formulated as the

Conjecture.  If  {x1,x2,,xn}  is multiplicatively independent, then the transcendence degreeMathworldPlanetmath of the set


is at least n.


  • 1 Diego Marques & Jonathan Sondow: Schanuel’s conjecture and algebraic powers zw and wz with z and w transcendental (2011). Available
Title multiplicatively independent
Canonical name MultiplicativelyIndependent
Date of creation 2013-03-22 19:36:03
Last modified on 2013-03-22 19:36:03
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 6
Author pahio (2872)
Entry type Definition
Classification msc 11J85
Classification msc 12F05