Quadratic field

 ( Algebraic Number Theory ) 

In algebraic number theory, a quadratic field is an algebraic number field of degree two over $\backslash mathbf\{Q\}$, the .

Every such quadratic field is some $\backslash mathbf\{Q\}(\backslash sqrt\{d\})$ where $d$ is a (uniquely defined) square-free integer different from $0$ and $1$. If $d>0$, the corresponding quadratic field is called a real quadratic field, and, if , it is called an imaginary quadratic field or a complex quadratic field, corresponding to whether or not it is a Field extension of the field of the .

Quadratic fields have been studied in great depth, initially as part of the theory of binary quadratic forms. There remain some unsolved problems. The class number problem is particularly important.

---

Ring of integers

---

Discriminant For a nonzero square free integer $d$, the discriminant of the quadratic field $K\; =\; \backslash mathbf\{Q\}(\backslash sqrt\{d\})$ is $d$ if $d$ is congruent to $1$ modulo $4$, and otherwise $4d$. For example, if $d$ is $-1$, then $K$ is the field of Gaussian rationals and the discriminant is $-4$. The reason for such a distinction is that the ring of integers of $K$ is generated by $(1+\backslash sqrt\{d\})/2$ in the first case and by $\backslash sqrt\{d\}$ in the second case.

The set of discriminants of quadratic fields is exactly the set of fundamental discriminants (apart from $1$, which is a fundamental discriminant but not the discriminant of a quadratic field).

---

Prime factorization into ideals Any prime number $p$ gives rise to an ideal $p\backslash mathcal\{O\}\_K$ in the ring of integers $\backslash mathcal\{O\}\_K$ of a quadratic field $K$. In line with general theory of splitting of prime ideals in Galois extensions, this may be

$p$ is inert prime

$(p)$ is a prime ideal.

The quotient ring is the finite field with $p^2$ elements: $\backslash mathcal\{O\}\_K\; /\; p\backslash mathcal\{O\}\_K\; =\; \backslash mathbf\{F\}\_\{p^2\}$.

$p$ split prime

$(p)$ is a product of two distinct prime ideals of $\backslash mathcal\{O\}\_K$.

The quotient ring is the product $\backslash mathcal\{O\}\_K/p\backslash mathcal\{O\}\_K\; =\; \backslash mathbf\{F\}\_p\backslash times\backslash mathbf\{F\}\_p$.

$p$ is ramified prime

$(p)$ is the square of a prime ideal of $\backslash mathcal\{O\}\_K$.

The quotient ring contains non-zero nilpotent elements.

The third case happens if and only if $p$ divides the discriminant $D$. The first and second cases occur when the Kronecker symbol $(D/p)$ equals $-1$ and $+1$, respectively. For example, if $p$ is an odd prime not dividing $D$, then $p$ splits if and only if $D$ is congruent to a square modulo $p$. The first two cases are, in a certain sense, equally likely to occur as $p$ runs through the primes—see Chebotarev density theorem.

The law of quadratic reciprocity implies that the splitting behaviour of a prime $p$ in a quadratic field depends only on $p$ modulo $D$, where $D$ is the field discriminant.

---

Class group Determining the class group of a quadratic field extension can be accomplished using Minkowski's bound and the Kronecker symbol because of the finiteness of the class group. A quadratic field $K\; =\; \backslash mathbf\{Q\}(\backslash sqrt\{d\})$ has discriminant so the Minkowski bound is$$M\_K\; =\; \backslash begin\{cases\}\; 2\backslash sqrt$$/\pi & d < 0 \\ \sqrt/2 & d > 0 . \end{cases}

Then, the ideal class group is generated by the prime ideals whose norm is less than $M\_K$. This can be done by looking at the decomposition of the ideals $(p)$ for $p\; \backslash in\; \backslash mathbf\{Z\}$ prime where ^{page 72} These decompositions can be found using the Dedekind–Kummer theorem.

---

Quadratic subfields of cyclotomic fields

---

The quadratic subfield of the prime cyclotomic field A classical example of the construction of a quadratic field is to take the unique quadratic field inside the cyclotomic field generated by a primitive $p$th root of unity, with $p$ an odd prime number. The uniqueness is a consequence of Galois theory, there being a unique subgroup of index $2$ in the Galois group over $\backslash mathbf\{Q\}$. As explained at Gaussian period, the discriminant of the quadratic field is $p$ for $p=4n+1$ and $-p$ for $p=4n+3$. This can also be predicted from enough ramification theory. In fact, $p$ is the only prime that ramifies in the cyclotomic field, so $p$ is the only prime that can divide the quadratic field discriminant. That rules out the 'other' discriminants $-4p$ and $4p$ in the respective cases.

---

Other cyclotomic fields If one takes the other cyclotomic fields, they have Galois groups with extra $2$-torsion, so contain at least three quadratic fields. In general a quadratic field of field discriminant $D$ can be obtained as a subfield of a cyclotomic field of $D$-th roots of unity. This expresses the fact that the conductor of a quadratic field is the absolute value of its discriminant, a special case of the conductor-discriminant formula.

---

Orders of quadratic number fields of small discriminant The following table shows some orders of small discriminant of quadratic fields. The maximal order of an algebraic number field is its ring of integers, and the discriminant of the maximal order is the discriminant of the field. The discriminant of a non-maximal order is the product of the discriminant of the corresponding maximal order by the square of the determinant of the matrix that expresses a basis of the non-maximal order over a basis of the maximal order. All these discriminants may be defined by the formula of .

For real quadratic integer rings, the ideal class number, which measures the failure of unique factorization, is given in OEIS A003649; for the imaginary case, they are given in OEIS A000924.

$\backslash mathbf\{Z\}\backslash left\backslash sqrt\{-5\}\backslash right$	$-20$	$2$	$\backslash pm\; 1$	Ideal classes $(1)$, $(2,1+\backslash sqrt\{-5\})$
$\backslash mathbf\{Z\}\backslash left\backslash tfrac\{1\}\{2\}(1+\backslash sqrt\{-19\})\backslash right$	$-19$	$1$	$\backslash pm\; 1$	Principal ideal domain, not Euclidean domain
$\backslash mathbf\{Z\}\backslash left2\backslash sqrt\{-1\}\backslash right$	$-16$	$1$	$\backslash pm\; 1$	Non-maximal order
$\backslash mathbf\{Z\}\backslash left\backslash tfrac\{1\}\{2\}(1+\backslash sqrt\{-15\})\backslash right$	$-15$	$2$	$\backslash pm\; 1$	Ideal classes $(1)$, $\backslash left(1,\backslash tfrac\{1\}\{2\}(1+\backslash sqrt\{-15\})\backslash right)$
$\backslash mathbf\{Z\}\backslash left\backslash sqrt\{-3\}\backslash right$	$-12$	$1$	$\backslash pm\; 1$	Non-maximal order
$\backslash mathbf\{Z\}\backslash left\backslash tfrac\{1\}\{2\}(1+\backslash sqrt\{-11\})\backslash right$	$-11$	$1$	$\backslash pm\; 1$	Euclidean
$\backslash mathbf\{Z\}\backslash left\backslash sqrt\{-2\}\backslash right$	$-8$	$1$	$\backslash pm\; 1$	Euclidean
$\backslash mathbf\{Z\}\backslash left\backslash tfrac\{1\}\{2\}(1+\backslash sqrt\{-7\})\backslash right$	$-7$	$1$	$\backslash pm\; 1$	Kleinian integers
$\backslash mathbf\{Z\}\backslash left\backslash sqrt\{-1\}\backslash right$	$-4$	$1$	$\backslash pm\; 1,\backslash pm\; i$ (cyclic of order $4$)	Gaussian integers
$\backslash mathbf\{Z\}\backslash left\backslash tfrac\{1\}\{2\}(1+\backslash sqrt\{-3\})\backslash right$	$-3$	$1$	$\backslash pm\; 1,\backslash tfrac\{1\}\{2\}(\backslash pm\; 1\; \backslash pm\; \backslash sqrt\{-3\})$ (cyclic of order $6$)	Eisenstein integers
$\backslash mathbf\{Z\}\backslash left$	$-84$	$4$		Class group non-cyclic: $(\backslash mathbf\{Z\}/2\backslash mathbf\{Z\})^2$
$\backslash mathbf\{Z\}\backslash left$	$5$	$1$	$\backslash pm\backslash left(\backslash tfrac\{1\}\{2\}(1+\backslash sqrt\{5\})\backslash right)^n$ (norm $(-1)^n$)	Golden field
$\backslash mathbf\{Z\}\backslash left$	$8$	$1$	$\backslash pm(1+\backslash sqrt\{2\})^n$ (norm $(-1)^n$)
$\backslash mathbf\{Z\}\backslash left$	$12$	$1$	$\backslash pm(2+\backslash sqrt\{3\})^n$ (norm $1$)
$\backslash mathbf\{Z\}\backslash left$	$13$	$1$	$\backslash pm\; \backslash left(\backslash tfrac\{1\}\{2\}(3+\backslash sqrt\{13\})\backslash right)^n$ (norm $(-1)^n$)
$\backslash mathbf\{Z\}\backslash left$	$17$	$1$	$\backslash pm(4+\backslash sqrt\{17\})^n$ (norm $(-1)^n$)
$\backslash mathbf\{Z\}\backslash left\backslash sqrt\{5\}\backslash right$	$20$	$1$	$\backslash pm(\backslash sqrt\{5\}+2)^n$ (norm $(-1)^n$)	Non-maximal order

Some of these examples are listed in Artin, Algebra (2nd ed.), §13.8.

---

See also

• Eisenstein–Kronecker number
• Genus character
• Heegner number
• Infrastructure (number theory)
• Quadratic integer
• Quadratic irrational
• Stark–Heegner theorem
• Dedekind zeta function
• Quadratically closed field

---

Notes

• Chapter 6.
• Chapter 3.1.

---

External links

Post Comment