In this paper we prove that solutions of q-P (A 7 ) are all transcendental over C(t).We also investigate transcendence of solutions of q-P (A 6 ) and prove transcendence of hypergeometric solutions of q-P (A 6 ).

Abstract. In this paper, we will study a property of solutions of q-Painlevé equation of type A ð1Þ 7 . We propose the notion of decomposable extension and then prove the equation has no solution in any decomposable extension of CðtÞ.

Abstract. We define the decomposable extensions of di¤erence fields and study the irreducibility of q-Painlevé equation of type A ð1Þ 0 7 . Every strongly normal extension or Liouville-Franke extension, the latter of which is a di¤erence analogue of the Liouvillian extension, satisfies that its appropriate algebraic closure is a decomposable extension.

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