  Multiple zeta values associated with function fields with varying constant
fields are dealt with simultaneously. Thakur introduced multiple zeta values in
the arithmetic of positive characteristic function fields, and the definition
depends on the field of constants of the chosen function field. Using
Papanikolas' theory on the relationship between the $t$-motivic Galois group
and the periods of a pre-$t$-motive, we show that there exist no algebraic
relations which relate multiple zeta values with different constants field.
