Quadratic Field

An Algebraic Integer of the form $a+b\sqrt{D}$ where $D$ is Squarefree forms a quadratic field and is denoted $\Bbb{Q}(\sqrt{D}\,)$. If $D>0$, the field is called a Real Quadratic Field, and if $D<0$, it is called an Imaginary Quadratic Field. The integers in ${\Bbb{Q}}(\sqrt{1}\,)$ are simply called ``the'' Integers. The integers in ${\Bbb{Q}}(\sqrt{-1}\,)$ are called Gaussian Integers, and the integers in ${\Bbb{Q}}(\sqrt{-3}\,)$ are called Eisenstein Integers. The Algebraic Integers in an arbitrary quadratic field do not necessarily have unique factorizations. For example, the fields $\Bbb{Q}(\sqrt{-5}\,)$ and $\Bbb{Q}(\sqrt{-6}\,)$ are not uniquely factorable, since

$$21=3\cdot 7=(1+2\sqrt{-5}\,)(1-2\sqrt{-5}\,)$$ (1)

$$6=-\sqrt{6}\,(\sqrt{-6}\,)=2\cdot 3,$$ (2)

although the above factors are all primes within these fields. All other quadratic fields $\Bbb{Q}(\sqrt{D}\,)$ with $\vert D\vert\leq 7$ are uniquely factorable.

Quadratic fields obey the identities

$$(a+b\sqrt{D})\pm(c+d\sqrt{D})=(a\pm c)+(b\pm d)\sqrt{D},$$ (3)

$$(a+b\sqrt{D})(c+d\sqrt{D})=(ac+bdD)+(ad+bc)\sqrt{D},$$ (4)

and

$${a+b\sqrt{D}\over c+d\sqrt{D}} = {ac-bdD\over c^2-d^2D} +{bc-ad\over c^2-d^2D} \sqrt{D}.$$ (5)

The Integers in the real field $\Bbb{Q}(\sqrt{D}\,)$ are of the form $r+s\rho$, where

$$\rho=\cases{ \sqrt{D} & for $D\equiv 2$\ or $D\equiv 3$\ (m... ...tstyle{1\over 2}}(-1+\sqrt{D}) & for $D\equiv 1$\ (mod 4).\cr}$$ (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.