Math 249 A Fall 2010 : Transcendental Number Theory
نویسندگان
چکیده
α is algebraic if there exists p ∈ Z[x], p 6= 0 with p(α) = 0, otherwise α is called transcendental . Cantor: Algebraic numbers are countable, so transcendental numbers exist, and are a measure 1 set in [0, 1], but it is hard to prove transcendence for any particular number. Examples of (proported) transcendental numbers: e, π, γ, e, √ 2 √ 2 , ζ(3), ζ(5) . . . Know: e, π, e, √ 2 √ 2 are transcendental. We don’t even know if γ and ζ(5), ζ(7), . . . are irrational or rational, and we know that ζ(3) is irrational, but not whether or not it is transcendental! Lioville showed that the number
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