page_title: Quadratic integer
section_title: Examples of complex quadratic integer rings
hierarchical_section_title: Quadratic integer / Quadratic integer rings / Examples of complex quadratic integer rings
context_page_description: In number theory, quadratic integers are a generalization of the integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form
  x² + bx + c = 0
  with b and c integers. When algebraic integers are considered, the usual integers are often called rational integers.
  Common examples of quadratic integers are the square roots of integers, such as √2, and the complex number i = √–1, which generates the Gaussian integers. Another common example is the non-real cubic root of unity, which generates the Eisenstein integers.
  Quadratic integers occur in the solutions of many Diophantine equations, such as Pell's equations, and other questions related to integral quadratic forms. The study of rings of quadratic integers is basic for many questions of algebraic number theory.
context_section_description: For D < 0, ω is a complex (imaginary or otherwise non-real) number. Therefore, it is natural to treat a quadratic integer ring as a set of algebraic complex numbers.
  A classic example is , the Gaussian integers, which was introduced by Carl Gauss around 1800 to state his biquadratic reciprocity law.
  The elements in  are called Eisenstein integers.
  Both rings mentioned above are rings of integers of cyclotomic fields Q(ζ₄) and Q(ζ₃) correspondingly.
  In contrast, Z[√−3] is not even a Dedekind domain.
  Both above examples are principal ideal rings and also Euclidean domains for the norm. This is not the case for 
  which is not even a unique factorization domain. This can be shown as follows.
  In  we have
  The factors 3,  and  are irreducible, as they have all a norm of 9, and if they were not irreducible, they would have a factor of norm 3, which is impossible, the norm of an element different of ±1 being at least 4. Thus the factorization of 9 into irreducible factors is not unique.
  The ideals  and  are not principal, as a simple computation shows that their product is the ideal generated by 3, and, if they were principal, this would imply that 3 would not be irreducible.
caption: Gaussian integers