In Mathematical, a quadratic body is a Extension of body $\backslash \; mathbb\; \{K\}\; \backslash \; backslash\; \backslash \; mathbb\; \{Q\}\; \backslash ,$ of the form

$K=\; \backslash \; mathbb\; \{Q\}\; (\backslash \; sqrt\; \{D\})$

where D is a rational Nombre different from zero. Such extensions cover all the extensions (body) of the body of the rational numbers which are of degree 2 (quadratic extensions). If D > 0, this one is called a real quadratic body , and for D < a 0 imaginary quadratic body . Such bodies are a class of basic examples in Algebraic theory of the numbers: the imaginary quadratic bodies are of an private interest, because they are the only bodies of algebraic numbers, outre$\backslash \; mathbb\; \{Q\}$, whose Groupe of the units is finished, and thus have arithmetic the simpler. They were studied very deeply, initially like part of the theory of the quadratic forms. There remain some unsolved problems.

One can limit oneself if D is a Integer, and makes some, D can be multiplied by any Square perfect without the extension changing; thus, to obtain all the bodies exactly once, one chooses a D representative. It is thus possible to take D as being an integer Without square, positive or negative. The Discriminant of the quadratic body correspondent is then D , if D is adequate to 1 modulo 4, and differently, 4 D .

For example, when D is - 1, K is the body of the rational of Gauss, the discriminant is - 4. The reason of this definition is related to the algebraic theory of the numbers (in the general case); the algebraic whole in K are generated by 1 and the square Racine of D only in the second case, whereas in the first case, there exist other algebraic entireties (form $(a+b\; \backslash \; sqrt\; d)/2$), for example, when D = - 3, the complex cubic roots of the unit.

The quadratic subfield of the cyclotomic body first

A traditional example of construction of a quadratic body is to take the single quadratic body inside the cyclotomic Corps generated by a primitive root p - ième of the unit, with p , a Prime number > 2. Unicity is a consequence of the Théorie of Welshman, being a single sub-group of index 2 in the group of Welsh on $\backslash \; mathbb\; \{Q\}\; \backslash ,$. As explained in the article of the Period of Gauss, the discriminant of the quadratic body is p for $p\; =\; 4n\; +\; 1\; \backslash ,$ and - p for $p\; =\; 4n\; +\; 3\; \backslash ,$. This can also be predicted starting from the theory of the Ramification S. In fact, p is the only prime number which ramifies in the cyclotomic body, i.e. p is the only prime number which can divide the discriminant of the quadratic body. This excluded the “others” disciminants -4p and 4p in the respective cases.

If the other cyclotomic bodies are taken, they have groups of Welshman with additional 2-torsions, and thus contain at least three quadratic bodies.

Decomposition in factors first in the ideals

Any prime number p generates an ideal $p.O\_K\; \backslash ,$ in the Anneau of the entireties $O\_K\; \backslash ,$ of a quadratic body $\backslash \; mathbb\; \{K\}\; \backslash ,$. According to the general theory, this can be

1. an ideal first, or

2. a product of two ideals first distinct from $O\_K\; \backslash ,$, or
3. the square of an ideal first of $O\_K\; \backslash ,$.

The third case appears only for the prime numbers dividing the discriminant. The two other cases appear both, when p varies, and in a certain direction are similar. One says in the first case that p remains inert , and in the second case that p is broken up ; if p is inert , the ring-quotient $O\_K\; \backslash \; backslash\; p.O\_K\; \backslash ,$ is a Corps finished having $p^2\; \backslash ,$ elements, if p is broken up , it is a product of two finished bodies (isomorphous) having each one p elements. In the third case, one says that p ramifies , and the ring-quotient contains elements Nilpotent S different from zero.

See too

• Theorem of Stark-Heegner

• Number of quadratic Heegner
• irrational Number