First Sentence:
We start with a brief survey of the basic directions of research in the theory of transcendental numbers.
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Key Phrases - Statistically Improbable Phrases (SIPs):
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algebraic independence measure, algebraic number field containing, linearly irreducible, exact denominators, linearly reducible, order algebraic differential equation, arbitrary algebraic equations, transcendence measure, homogeneous linear differential equations, certain linear forms, approximating forms, minimal equations, order homogeneous linear differential equation, transcendental number theory, transcendence degree, single algebraic equation, power series coefficients, auxiliary theorem, suppose that the theorem, algebraic coefficients, arithmetic properties, algebraic points, first differential equation, homogeneous algebraic equation, irreducibility condition
Key Phrases - Capitalized Phrases (CAPs):
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Second Fundamental Theorem, First Fundamental Theorem, Third Fundamental Theorem, Kronecker's Theorem, Proof Let
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