section_path: Applications/Algebraic number theory
section_name: Algebraic number theory
paragraph_text: Algebraic number theory
  
  As mentioned above, any nonconstant polynomial equation (in complex coefficients) has a solution in C {\displaystyle \mathbb {C} }. A fortiori, the same is true if the equation has rational coefficients. The roots of such equations are called algebraic numbers – they are a principal object of study in algebraic number theory. Compared to Q ¯ {\displaystyle {\overline {\mathbb {Q} }}}, the algebraic closure of Q {\displaystyle \mathbb {Q} }, which also contains all algebraic numbers, C {\displaystyle \mathbb {C} } has the advantage of being easily understandable in geometric terms. In this way, algebraic methods can be used to study geometric questions and vice versa. With algebraic methods, more specifically applying the machinery of field theory to the number field containing roots of unity, it can be shown that it is not possible to construct a regular nonagon using only compass and straightedge – a purely geometric problem.
  
  Another example is the Gaussian integers; that is, numbers of the form x + iy, where x and y are integers, which can be used to classify sums of squares.