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R ∈ 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. By substituting αi for Xi for i = 1, .

11) is false. 8 is false leads to a contradiction. 12. We follow the transcendence proof of e, with the necessary modifications. Before proceeding, we observe that there is no loss of generality to assume that δ1 , . . , δt are algebraic integers. Indeed, there is a positive m ∈ Z such that mδ1 , . . g, we may take for m the product of the denominators of β1 , . . 12 are unaffected if we replace δi by mδi for i = 1, . . , t. Let γ1 , . . , γt be distinct algebraic numbers and δ1 , . . , δt non-zero algebraic integers from the normal number field L, such that each τ ∈ Gal(L/Q) permutes the pairs (γ1 , δ1 ), .

Then C is finite over A. Proof. Suppose that B is generated as an A-module by α1 , . . , αm , and C is generated as a B-module by β1 , . . , βn . A straightforward computation shows that C is generated an as A-module by αi βj (i = 1, . . , m, j = 1, . . , n). 14. Let A, B be commutative rings. Then B is finite over A if and only if B = A[α1 , . . , αr ] for certain α1 , . . , αr that are integral over A. Proof. Suppose that B is finite over A, say B is generated as an A-module by α1 , . .