although the above factors are all primes within these fields. All other quadratic fields
with are
uniquely factorable.
Quadratic fields obey the identities
(3)
(4)
and
(5)
The integers in the real field are of the form , where
(6)
There are exactly 21 quadratic fields in which there is a Euclidean algorithm, corresponding to for squarefree integers
, , , , , 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, and
73 (A048981). This list was published by Inkeri
(1947), but erroneously included the spurious additional term 97 (Barnes and Swinnerton-Dyer
1952; Hardy and Wright 1979, p. 217).
Barnes, E. S. and Swinnerton-Dyer, H. P. F. "The Inhomogeneous Minima of Binary Quadratic Forms. I." Acta Math87,
259-323, 1952.Berg, E. Fysiogr. Sällsk. Lund. Föhr.5,
1-6, 1935.Chatland, H. "On the Euclidean Algorithm in Quadratic
Number Fields." Bull. Amer. Math. Soc.55, 948-953, 1949.Chatland,
H. and Davenport, H. "Euclid's Algorithm in Real Quadratic Fields." Canad.
J. Math.2, 289-296, 1950.Hardy, G. H. and Wright, E. M.
"Real Euclidean Fields" and "Real Euclidean Fields (Continued)."
§14.8 and 14.9 in An
Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon
Press, pp. 213-217, 1979.Inkeri, K. "Über den Euklidischen
Algorithmus in quadratischen Zahlkörpern." Ann. Acad. Sci. Fennicae
Ser. A. 1. Math.-Phys., No. 41, 1-35, 1947.Koch, H. "Quadratic
Number Fields." Ch. 9 in Number
Theory: Algebraic Numbers and Functions. Providence, RI: Amer. Math. Soc.,
pp. 275-314, 2000.LeVeque, W. J. Topics
in Number Theory, Vol. 2. Reading, MA: Addison-Wesley, p. 57, 1956.Oppenheim.
Math. Ann.109, 349-352, 1934.Samuel, P. "Unique
Factorization." Amer. Math. Monthly75, 945-952, 1968.Stark,
H. M. An
Introduction to Number Theory. Cambridge, MA: MIT Press, p. 294, 1994.Shanks,
D. Solved
and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 153-154,
1993.Sloane, N. J. A. Sequence A048981
in "The On-Line Encyclopedia of Integer Sequences."
where 
is squarefree forms a quadratic field and is denoted
. If 
, the field is called a real
quadratic field, and if 
, it is called an imaginary
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. The algebraic
integers in an arbitrary quadratic field do not necessarily have unique factorizations.
For example, the fields 
and 
are not uniquely factorable, since
with 
are
uniquely factorable.
are of the form 
, where
for squarefree integers
, 
, 
, 
, 
, 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, and
73 (A048981). This list was published by Inkeri
(1947), but erroneously included the spurious additional term 97 (Barnes and Swinnerton-Dyer
1952; Hardy and Wright 1979, p. 217).
