A quadratic integer is an algebraic integer in a quadratic number field , where is a squarefree integer other than . Its ring of integers is if , and otherwise. Thus is a quadratic integer, since it satisfies the monic equation .
For , its conjugate is . Its field trace and field norm are and ; their integrality characterizes the algebraic integers in this degree-two extension. These formulas connect Diophantine equations to arithmetic in a larger ring.
Articles by others on the same topic
A quadratic integer is a type of algebraic integer that is a root of a monic polynomial of degree two with integer coefficients. In simpler terms, a quadratic integer can be expressed in the form \( a + b\sqrt{d} \), where \( a \) and \( b \) are integers, and \( d \) is a square-free integer (i.e., \( d \) is not divisible by the square of any prime).