# Proving the set of quadratic numbers is countable

I don't really know where to start with this, thought I'd ask for some input. Here's the question:

Prove that the set of quadratic numbers is countable.

EDIT The question says that a number is quadratic if it is a root of a degree two polynomial.

• What is your definition of a "quadratic number?" I can think of a few... – Thomas Andrews Sep 20 '17 at 0:28
• Are you asking about square numbers like $1,4,9,16,25...$? – AlgorithmsX Sep 20 '17 at 0:37
• The question says that a number is quadratic if it is a root of a degree two polynomial. – Boo92 Sep 20 '17 at 0:42
• The coefficients of the quadratic polynomial determine the nature of its roots. If the coefficients are rational, then the quadratic numbers are countable. If the coefficients are complex, they are not. – N. F. Taussig Sep 20 '17 at 10:50

## 1 Answer

In my mind, i reworded your question as "is the number of quadratic (degree 2) extensions of Q countable ?"
This nicely restricts us to just (degree two) polynomials with rational coefficients, as clearly the complex or real case is uncountable. This is then a subset of all polynomials with rational coefficients, which is countable (algebraic integers are countable).