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A number x0 ∈ R is called algebraic of degree n if it is the root of a polynomial P(x) = anxn + ............... + a1x + a0, where aj ∈ Z, an ≠ 0, and n is minimal. A number x0 that is not algebraic is called transcendental.
a) Prove that if n ∈ N and q ∈ Q, then nq is algebraic.
b) Prove that for each n ∈ N the collection of algebraic numbers of degree n is countable.
c) Prove that the collection of transcendental numbers is uncountable. (Two famous transcendental numbers are n and e. For more information on transcendental numbers and their history, see Kline [5].) |
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