Dryope

See also: Body

In Mathematical, and more precisely in Algebra, a body is a algebraic Structure. In an abstract way, a body is a unit in which it is possible to carry out additions, subtractions, multiplications and divisions.

Elementary examples of body are the body of the rational numbers (noted $\backslash \; mathbb\; \{Q\}$), the body of the real numbers (noted $\backslash \; mathbb\; \{R\}$), the body of the complex numbers (noted $\backslash \; mathbb\; \{C\}$) and the body $\backslash \; mathbb\; \{F\}\; \_p$ of congruences modulo p where p is a prime number. The simplest example of noncommutative body (sometimes called ring with division , according to English) is that of the Quaternion S.

The theory of the bodies is called, by some, Théorie of Welshman; however, the Welshman theory in general indicates a method of study which applies in particular to the commutative bodies and the extensions of bodies, which form the historical example, but also extends to good from other fields, for example the study of the differential equations (differential Théorie of Welshman), or of the coatings. In other words, the Welshman theory is a branch of the theory of the bodies.

Fragments of history

Until the 19th century, the Ensemble S of numbers appeared so natural that one was never concerned with give them a name, nor to even define with precision their structure. However, with the birth of the study of the algebraic numbers, it appeared other whole of numbers that the rational ones, realities and the complexes. It became necessary to specify the structure of body, then the concept of entireties on this body and finally the concept of ring. It is at the German school that one owes the development of these concepts. It is Richard Dedekind which defines for the first time the structure of body (German Körper) and this is why an unspecified body is often named K . The structure of body forms part of a hierarchy including/understanding the Monoïde, the group, the ring, and gives place to the definition of the vector Space, and the algebra.

Definition and example

A body is a unit K provided with two internal laws noted in general + and × checking

• ( K , +) form a commutative group whose neutral element is noted 0
• ( K - {0}, ×) form a multiplicative group.
• the multiplication is distributive for the addition (on the left as on the right) i.e.

$\backslash \; forall\; (has,\; B,\; c)\; \backslash \; in\; K^3,\; \backslash \; quad\; has\; \backslash \; times\; (B\; +\; c)\; =\; has\; \backslash \; times\; B\; +\; has\; \backslash \; times\; C\; \backslash \; quad\; \backslash \; hbox\; \{and\}\; \backslash \; quad\; (b+\; c)\; \backslash \; times\; has\; =\; B\; \backslash \; times\; has\; +\; C\; \backslash \; times\; a$

One then speaks about the body (K, +, ×)

The first studied bodies being whole of numbers (rational, real, complex, algebraic), the multiplication were commutative there. This is why, initially, in the definition of a body, the multiplication was to be commutative. Currently, the tendency is rather not to require the commutation of the multiplication and to specify commutative body if necessary. The noncommutative bodies are sometimes called left bodies or rings with division or. This terminology is inspired by English where a commutative body is called field and a body not necessarily commutative division boxing ring .

Examples of body

• the whole of the rational numbers, $(\backslash \; mathbb\; Q,\; +,\; \backslash \; times)$ is a commutative body
• the whole of the real numbers $(\backslash \; mathbb\; R,\; +,\; \backslash \; times)$ is a commutative body
• the whole of the complex $(\backslash \; mathbb\; C,\; +,\; \backslash \; times)$ is a commutative body
• the whole of the Quaternion S $(\backslash \; mathbb\; H,\; +,\; \backslash \; times)$ is a noncommutative body
• In arithmetic modular, the unit $(\backslash \; mathbb\; Z/p\; \backslash mathbb\; Z,\; +,\; \backslash \; times)$ where p is a prime number is a commutative body

A subfield of a body K is a part nonempty L of K, stable by + and $\backslash \; times$, such as L provided with the induced laws is a body.

Characteristic

See also: Characteristic of a ring

If there exists a natural entirety N not no one such as $1\; +\; 1\; +\; \backslash \; cdots\; +\; 1$ (with N terms) is null, one calls characteristic of the body the smallest positive entirety not no one checking this property. If there does not exist entirety not no one checking this property, it is said that the body is of null characteristic (sometimes as large as one wants).

For example the body $\backslash \; R$ is of null characteristic whereas the body $(\backslash \; mathbb\; Z/p\; \backslash \; mathbb\; Z)$ is of characteristic p . It is shown that a body always has for characteristic either a 0 or prime number.

Finished bodies

See also: Body finished

These are the bodies of which the number of elements is finished. The study of the finished bodies is late in the study of the bodies. One shows that a finished body is always commutative, of cardinal equal to the power of a prime number. It is in fact possible to draw up the list of all the finished bodies (except for isomorphism).

The smallest finished body is that of the Boolean ones, of which here tables of addition and multiplication:

The finished bodies most known are the bodies of congruences modulo a prime number as in the case above, but there is an infinity of others, such as for example those, respectively to four and nine elements, of which we give below the “tables of Pythagore”, successively for the first law of composition known as “addition”, then for the second known as “multiplication”. We indicate in each case as with the neutral element of the first law of composition, B that of the second.

Four elements:

Last nine elements:

Body and ring

The unit $(\backslash \; mathbb\; Z,\; +,\; \backslash \; times)$ is not a body because the majority of the elements of $\backslash \; mathbb\; Z^*$ are not invertible: for example, there thus does not exist relative entirety N such as 2n = 1 2 is not invertible.

More generally, a unit has provided with two laws + and × checking

• (has, +) form a commutative group whose neutral element is noted 0
• (Has {0}, ×) form a Monoïde.
• the multiplication is distributive for the addition (on the left like on the right)

is a unit ring

See also: Ring (mathematics)

If the ring is commutative and just, i.e. if

$\backslash \; forall\; (has,\; b)\; \backslash \; in\; A^2,\; \backslash \; quad\; ab=0\; \backslash \; Rightarrow\; (a=0\; \backslash \; hbox\; \{or\}\; b=0)$,

or, $\backslash \; forall\; (has,\; b)\; \backslash \; in\; A^2,\; \backslash \; quad\; (ab=0\; \backslash \; hbox\; \{and\}\; has\; \backslash \; neq0)\; \backslash \; Rightarrow\; b=0$

the ring is almost a body because it misses nothing any more but the inversibility for the multiplication. It is shown that one can plunge the ring in his body of the fractions, which is the smallest body containing the ring.

See also: Body of the fractions

Example: $\backslash \; mathbb\; Q$ is the body of the fractions of $\backslash \; mathbb\; Z$

Body and vector space

See also: vector Space

On the basis of the body $\backslash \; R$, it is natural to be interested à$\backslash \; R^n$, together tuple of realities. One is brought to provide it with an addition and a multiplication by a reality. The structure thus defined (an internal addition providing the unit with a structure of group and an external multiplication having of the properties of distributivity and associativeness) is called vector space on $\backslash \; R$. It is then natural to define what could be a vector space on an unspecified body K.

Body and algebraic equation

The study of the Polynomial S with coefficient in a commutative body and the search for their roots developed the concept of body considerably. If F is a polynomial of degree N on a commutative body K, the equation F (X) = 0 is an algebraic equation in K. If, moreover, F is an irreducible polynomial, the equation is known as irreducible. When N is equal to or higher than two, to find the solutions of a such equation requires to be placed in a body larger than K, a Extension of body.

For example, the equation $x^2-2\; =\; 0$ is irreducible in $\backslash \; mathbb\; Q$ but has roots dans$\backslash \; mathbb\; R$ or better in $\backslash \; mathbb\; Q\; 2$. The equation $x^2+1\; =\; 0$ does not have a solution in $\backslash \; mathbb\; R$ but has some in $\backslash \; mathbb\; C$ or better in $\backslash \; mathbb\; Q$.

A Corps of rupture of a polynomial is, for example, a minimal body containing K and a root of F .

The Corps of decomposition of F is the smallest body containing K like all the roots of F .

The study of the bodies of decomposition of a polynomial and group of Permutation of its roots forms the branch of mathematics which one calls the Théorie of Welshman.

See also: Extension of body, algebraic Extension, Theory of Welshman

Other fields of study

One finds the theory of the bodies in the study of certain functions like the rational functions or the elliptic functions

Additional structures

• Body valué
• Body ordered

See too

Random links:

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