Recitation 17

In section D, I proved that the set of all algebraic numbers is countable.

An algebraic number is a real number that is one of the roots of a polynomial with rational coefficients. For example, all rational are algebraic. Also, the square root of a rational is algebraic.

Let P denote the set of all polynomials with rational coefficients and P_n be the set of all such polynomials with degree \leq n. Because there is a bijection between P_n and the cartesian product of n many \mathbb{Q}‘s, P_n is countable.

Since P=\bigcup_{n=0}^\infty P_n, P is countable as well. Notice that each polynomial has only finte roots. As P is countable, the roots generated by the polynomials in P are countable.

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