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Quadratic Field

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A quadratic field is a number field of extension degree 2 over the rational numbers. It has the form Q(sqrt(D))={a+bsqrt(D):a,b in Q}, where D is a nonzero squarefree integer different from 1. If D>0, it is a real quadratic field, and if D<0, it is an imaginary quadratic field. The ring of integers of Q is Z. The ring of integers of Q(sqrt(-1)) consists of the Gaussian integers, and that of Q(sqrt(-3)) consists of the Eisenstein integers. The ring of integers of a quadratic field need not be a unique factorization domain. For example, the rings of integers of Q(sqrt(-5)) and Q(sqrt(-6)) have the distinct factorizations

21=3·7=(1+2sqrt(-5))(1-2sqrt(-5))
(1)
6=-sqrt(-6)(sqrt(-6))=2·3,
(2)

whose factors are irreducible elements in the respective rings of integers. The rings of integers of all other quadratic fields Q(sqrt(D)) with |D|<=7 are unique factorization domains.

For rational numbers a, b, c, and d, the field operations satisfy

(a+bsqrt(D))+/-(c+dsqrt(D))=(a+/-c)+(b+/-d)sqrt(D)
(3)
(a+bsqrt(D))(c+dsqrt(D))=(ac+bdD)+(ad+bc)sqrt(D),
(4)

and

(a+bsqrt(D))/(c+dsqrt(D))=(ac-bdD)/(c^2-d^2D)+(bc-ad)/(c^2-d^2D)sqrt(D)
(5)

when c+dsqrt(D)!=0.

The ring of integers of Q(sqrt(D)) consists of r+srho with r,s in Z, where

rho={sqrt(D) for D=2 or D=3 (mod 4); 1/2(-1+sqrt(D)) for D=1 (mod 4).
(6)

There are exactly 21 quadratic fields whose rings of integers admit a Euclidean algorithm using |a^2-Db^2| as the Euclidean function for a+bsqrt(D). They are Q(sqrt(D)) for squarefree integers -11, -7, -3, -2, -1, 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, and 73 (OEIS 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).

See also

Algebraic Integer, Eisenstein Integer, Gaussian Integer, Imaginary Quadratic Field, Integer, Number Field, Quadratic, Real Quadratic Field

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References

Barnes, E. S. and Swinnerton-Dyer, H. P. F. "The Inhomogeneous Minima of Binary Quadratic Forms. I." Acta Math 87, 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. Monthly 75, 945-952, 1968.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."Stark, H. M. An Introduction to Number Theory. Cambridge, MA: MIT Press, p. 294, 1994.

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Quadratic Field

Cite this as:

Weisstein, Eric W. "Quadratic Field." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/QuadraticField.html

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