Quadratic Field
An Algebraic Integer of the form 
where 
is Squarefree forms a quadratic field and is denoted
. If 
, the field is called a Real Quadratic Field, and if 
, it is called anImaginary Quadratic Field. The integers in 
are simply called ``the''Integers. The integers in 
are called Gaussian Integers, and the integers in 
are called Eisenstein Integers. TheAlgebraic Integers in an arbitrary quadratic field do not necessarily have unique factorizations.For example, the fields 
and 
are not uniquely factorable, since
 | (1) |
 | (2) |
with
are uniquely factorable.Quadratic fields obey the identities
 | (3) |
 | (4) |
 | (5) |
The Integers in the real field are of theform 
, where
 | (6) |
There exist 22 quadratic fields in which there is a Euclidean Algorithm (Inkeri 1947).
See also Algebraic Integer, Eisenstein Integer, Gaussian Integer, Imaginary Quadratic Field,Integer, Number Field, Real Quadratic Field
References
Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 153-154, 1993.
- Orthobirotunda
- Orthocenter
- Orthocentric Coordinates
- Orthocentric Quadrilateral
- Orthocentric System
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- Orthodrome
- Orthogonal Array
- Orthogonal Basis
- Orthogonal Circles
- Orthogonal Curves
- Orthogonal Functions
- Orthogonal Group
- Orthogonal Group Representations
- Orthogonality Condition
- Orthogonality Theorem
- Orthogonal Lines
- Orthogonal Matrix
- Orthogonal Polynomials
- Orthogonal Projection
- Orthogonal Rotation Group
- Orthogonal Tensors
- Orthogonal Transformation
- Orthogonal Vectors
- Orthographic Projection