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Let K ⊂ L be a field extension and α1 , . . , αr ∈ L. Then α1 , . . , αr are algebraically dependent over K if and only if there is i such that αi is algebraic over K(α1 , . . , αi−1 , αi+1 , . . , αr ). Proof. Suppose that α1 , . . , αr are algebraically dependent over K. Then there is non-zero P ∈ K[X1 , . . , Xr ] with P (α1 , . . , αr ) = 0. Suppose that for instance the variable Xr occurs in P . Then we can write P as ti=0 Pi (X1 , . . , Xr−1 )Xri , where the Pi are polynomials with coefficients in K, with t > 0 and Pt = 0.

19. Let α1 , α2 , β1 , β2 ∈ Q be non-zero. Assume that log α1 , log α2 are linearly independent over Q. Then β1 log α1 + β2 log α2 = 0. Proof. Suppose β1 log α1 + β2 log α2 = 0. Put γ := −β2 /β1 . 17. In 1966, A. Baker proved the following far-reaching generalization. 20 (A. Baker, 1966). Let α1 , . . , αn , β1 , . . , βn ∈ Q be non-zero. Assume that log α1 , . . , log αn are linearly independent over Q. Then β1 log α1 + · · · + βn log αn is transcendental. Definition. We say that non-zero complex numbers α1 , .

Let A ⊂ B be an extension of commutative rings, and α ∈ B. Then the following are equivalent: (i) α is integral over A; (ii) A[α] is finite over A; (iii) there is a non-zero, finitely generated A-submodule M of B such that 1 ∈ M and αM ⊆ M , where αM = {αx : x ∈ M }. Proof. (i)=⇒(ii). Let f ∈ A[X] be a monic polynomial with f (α) = 0. Let β ∈ A[α]. Then β = g(α) with g ∈ A[X]. Since f is monic, using division with remainder we find q, r ∈ A[X] with g = qf + r, and deg r < deg f = n. We may write r = c0 + c1 X + · · · + cn−1 X n−1 with ci ∈ A.