In his book " Transcendent al and algebraic numbers" , [1] pp. 182-1 33, A. O. GJ~LFOND gives some results concerning the algebraic independence of numbers which can be considered as extensions of the fam ous t heore ms of Gelfond-Schne ider and Lindemann. More recently A. A. SHMELEV pu blished a variant of one of these theorems, [5]. H ere we shall prov e a generalizati on of t hese t hree theorems, viz .
T heore m 1. Suppose the numbers 1X0 , 1X1, 1X2 as well as the n umb ers 110, 111, 112 are linearl y independent over Q. Th en the extension of t he rational field by means of adj unct ion to it of the nu mbers k, j =0, 1, 2, will have tran scendence degree ~2.
T h eorem 2. Suppose the numbers C(ij, 1X1, 1X2 as well as the numbers "lo, 111 are linearly indep endent over Q. Then the ext ension of t he ra tiona l field by mean s of adj unct ion to it of the numbers k=O, 1,2 and j = O, 1, will have transcendence degree ~2.
The or em 3 . Suppose t he numbers 1X0, lXI, 1X2, 1X3 as well as the nu mbers "lo, 111 are linearl y independent over Q. Then the ex t ension of the ratio na l field by means of adjunction t o it of the numbers k=O, 1, 2,3 a nd j =O , 1, will have transcendence degree ~2.
GELl fOND, [1] Oh. III , Β§ Β§ 4, 5, has prove d theorem 1 under t he extra condition t hat for x lar ger than some x' the inequ ality i=O, 1, 2, hold s, where r > Β°is some constant and xo, Xl, X2 are rationa l integers, not all zer o. In the same way theorem 2 has been given by GELFOND, [1]