3.5. EXERCISES 75

9. Let V be an n dimensional vector space and let W be a subspace. Generalize theProblem 7 to define and give properties of V/W . What is its dimension? What is abasis?

10. A number is transcendental if it is not the root of any nonzero polynomial with ra-tional coefficients. As mentioned, there are many known transcendental numbers.Suppose α is a real transcendental number. Show that

{1,α,α2, · · ·

}is a linearly

independent set of real numbers if the field of scalars is the rational numbers.

11. Suppose F is a countable field and let A be the algebraic numbers, those numbers inG which are roots of a polynomial in F [x]. Show A is also countable.

12. It was shown in the chapter that A is a field. Here A are the numbers in R which areroots of a rational polynomial. Then it was shown in Problem 11 that it was actuallycountable. Show that A+ iA is also an example of a countable field.