Exercise 3, § 3

Show that every algebraic extension E of a field K is equipotent to a subset of $K \times N$ (consider the mapping which associates with each element of E its minimal polynomial over K). In particular, every algebraic extension of a finite field is countable and every algebraic extension of an infinite field K is equipotent to K. \* Deduce that in the field R of real numbers there exist numbers that are transcendental over the prime subfield $\mathbf{Q}$ and that the set of all these numbers has the power of the continuum.