Definition
Quadratic integers are solutions of equations of the form:
- x2 + Bx + C = 0
for integers B and C. Such solutions have the form a + ω b , where a, b are integers, and where ω is defined by:
(D is a square-free integer. Note that the case is impossible, since it would imply that D is divisible by 4, a perfect square, which contradicts the fact that D is square-free.).
This characterization was first given by Richard Dedekind in 1871.
The set of all quadratic integers is not closed even under addition. But for any fixed D the set of corresponding quadratic integers forms a ring, and it is these quadratic integer rings which are usually studied. Medieval Indian mathematicians had already discovered a multiplication of quadratic integers of the same D, which allows one to solve some cases of Pell's equation. The study of quadratic integers admits an algebraic version: the study of quadratic forms with integer coefficients.
Read more about this topic: Quadratic Integer
Famous quotes containing the word definition:
“Im beginning to think that the proper definition of Man is an animal that writes letters.”
—Lewis Carroll [Charles Lutwidge Dodgson] (18321898)
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (17721834)
“One definition of man is an intelligence served by organs.”
—Ralph Waldo Emerson (18031882)