An Introduction to Irrationality and Transcendence Methods. 3 Auxiliary Functions in Transcendence Proofs 3.1 Explicit Functions
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چکیده
This yields an irrationality criterion (which is the basic tool for proving the irrationality of specific numbers), and Liouville extended it into a transcendence criterion. The proof by Liouville involves the irreducible polynomial f ∈ Z[X] of the given irrational algebraic number α. Since α is algebraic, there exists an irreducible polynomial f ∈ Z[X] such that f(α) = 0. Let d be the degree of f . For p/q ∈ Q the number qf(p/q) is a non–zero rational integer, hence
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Auxiliary functions in transcendence proofs
We discuss the role of auxiliary functions in the development of transcendental number theory. Earlier auxiliary functions were completely explicit (§ 1). The earliest transcendence proof is due to Liouville (§ 1.1), who produced the first explicit examples of transcendental numbers at a time where their existence was not yet known; in his proof, the auxiliary function is just a polynomial in o...
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We discuss the role of auxiliary functions in the development of transcendental number theory. Earlier auxiliary functions were completely explicit (§ 1). The earliest transcendence proof is due to Liouville (§ 1.1), who produced the first explicit examples of transcendental numbers at a time where their existence was not yet known; in his proof, the auxiliary function is just a polynomial in o...
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