Body (algebra)

 Body affects the special fields mathematics is special case enclosure vector space as special cases

a body an excellent algebraic structure is in the mathematical subsection of algebra, in which the addition, subtraction, multiplication and division as with „the normal “(real) numbers can be accomplished.

The designation body became in 19. Century of smelling pool of broadcasting corporations Dedekind introduced. Bourbaki uses the terms differently: it calls inclined bodies body and body discussed here as commutative bodies.

formal definition

a commutative unitary ring, which is not the zero-ring, a body is called, if in it each element different of zero is multiplicatively invertable.

A body a commutative unitary ring is differently formulated [itex] K< /math>, inthat the group of units [itex] K^*< /math> equivalent [itex] K \ set minus \ {0 \}< /math>, thus maximally large is.

Here a specification of the necessary axioms:

A quantity [itex] (K, +, \ cdot)< /math> together with two binary linkages (those addition and multiplication to be usually called) is exactly a body if the following characteristics foreverything [itex] A, b, C \ in K< /math> are fulfilled:

1. [itex] a+ (b+c) = (a+b) +c< /math> (Assoziativität)
2. [itex] a+b = b+a< /math> (Kommutativität)
3. there is an element [itex] 0 \ in K< /math> with [itex] 0+a=a< /math> (neutral element)
4. to everyone [itex] A \ in K< /math> existed that additives Inverse [itex] - A< /math> with [itex] a+ (- A) =0< /math>
2. Multiplicative characteristics:
1. [itex] A \ cdot (b \ cdot C) = (A \ cdot b) \ cdot C< /math> (Assoziativität)
2. [itex] A \ cdot b = b \ cdot A< /math> (Kommutativität)
3. there is an element [itex] 1 \ in K< /math> with [itex] 1 \ a=a /math< cdot> (neutral element), and it is [itex] 1 \ ne0< /math>.
4. To everyone[itex] x \ in K \ set minus \ {0 \}< /math> exists the multiplicative inverse [itex] x^ {- to 1}< /math> with [itex] x \ cdot x^ {- 1} =1< /math>
3. Interaction of additiver and multiplicative structure:
1. [itex] A \ cdot (b+c) = A \ cdot b+a \ cdot C< /math> (Left Distributivität)

the right distributive law [itex] (a+b) \ cdot c=a \ cdot c+b \ cdot C< /math> follows then from the remaining characteristics.

Remarks

the definition it ensures for the fact that in a body addition and multiplication function in „the used “way:

• The inverse of [itex] A< /math> concerning the addition is [itex] - A< /math>, and becomes usually negatives of [itex] A< /math> or also „the additive inverse one “too [itex] A< /math>called.
• The inverse of [itex] A< /math> concerning the multiplication is [itex] a^ {- 1}< /math> and become the multiplicatively inverse or only inverse one or the reciprocal value of [itex] A< /math> called.
• <Math> 0< /math> the only element of the body, which does not have a reciprocal value, is the multiplicative group oneBody is thus [itex] K^ \ times=K \ set minus \ {0 \}< /math>.

Note:The formation negatives of the element does not have to do anything with the question whether the element is negative; for example that is negatives of the real number [itex] -2< /math> the positive number of 2. In a general bodythere is no term of negative or positive elements. (See body also arranged.)

characteristics

each body a ring is. The characteristics of the multiplicative group lift the body out from the rings. If the Kommutativität thatmultiplicative group, receives one is not demanded the term of the inclined body.

Each body is a vector space over itself (is called with itself as underlying scalar body).

examples

acquaintance of examples of bodies are the quantity that rational numbers [itex] (\ Bbb Q, +, \ cdot)< /math>, the quantity of the real numbers [itex] (\ R, +, \ cdot)< /math> and the quantity of the complex numbers [itex] (\ Bbb C, +, \ cdot)< /math> in each case with the usual addition and multiplication.

More complicated examples are finite bodies and the p-adischen numbers.

A counter example forms the quantitythe whole numbers [itex] (\ Bbb Z, +, \ cdot)< /math>: Is [itex] (\ Bbb Z, +)< /math> a group with neutral element [itex] 0< /math> and everyone [itex] A \ in \ Bbb Z< /math> that possesses additives inverse ones [itex] - A< /math>, but [itex] (\ Bbb {Z} \ set minus \ {0 \}, \ cdot)< /math> is no group. Nevertheless is [itex] 1< /math> the neutral element, but excepttoo [itex] 1< /math> and [itex] -1< /math> there are no multiplicative inverse ones (for example math <3^> is {- 1} = 1/3< /math> no whole, but a genuinly rational number). The whole numbers do not only form a body, but an integrity ring, whose quotient body is the rational numbers.

Unterkörper

an under and/or. Subfield is a subset of a body, which forms a body with the operations of the torso again. In addition the following statements for a Unterkörper must [itex] U< /math> a body [itex] K< /math> apply:

• [itex] A, b \ in U \ \ Rightarrow \ A +b \ in U, \ A \ cdot b \ in U< /math> (Compartmentation concerning addition and multiplication)
• [itex] 1_ {K}, \ 0_ {K} \ in U< /math> (The neutral elements of [itex] K< /math> A are <\> in< U> \
• < \> Rightarrow \ - A \ in U /math in math U /math)< math> (Each additives inverse one of [itex] U< /math> is in[itex] U< /math>)
• [itex] A \ in U \ set minus \ {0 \} \ \ Rightarrow \ a^ {- 1} \ in U< /math> (Each multiplicative inverse one of [itex] U< /math> (except that [itex] of 0< /math>) example is <in> math< U> /math

):

The body of the rational numbers [itex] \ mathbb {Q}< /math> a Unterkörper of the real numbers is [itex] \ R< /math>.

[itex] \ mathbb {Q}< /math> is even thatkleinst possible Unterkörper of [itex] \ R< /math>, i.e. each Unterkörper of [itex] \ R< /math> at least math <\> mathbb {Q contains}< of /math>. Each body of characteristic 0 somewhat more generally contains all whole numbers (1+1+···+1) and their inverse one, thus [itex] \ mathbb {Q}< /math>.

finite bodies

a body is in finite body, if its fundamental set [itex] K< /math> is finite. The finite bodies are completely classified in the following sense: Each finite body exactly math <q=p^n> /math< has> Elements with a prime number [itex] p< /math> and a positive natural number [itex] n< /math>. Up to isomorphism it givesto everyone such [itex] q=p^n< /math> exactly a finite body, that with [itex] \ Bbb F_q< /math> one designates. Each body [itex] \ Bbb F_ {p^n}< /math> the characteristic has [itex] p< /math>.

In the special case [itex] n=1< /math> we receive p /math to <each> prime number< math> the body [itex] \ Bbb F_p< /math>, that isomorphic to the remainder class ring< math>\ mathbb {Z} /p \ mathbb {Z}< /math> is.

Verallgemeinerung: Inclined body

major item: To inclined bodies

one does without the condition that the multiplication is commutative, then arrives one at the structure of the inclined body. There are however also authors, who presuppose for an inclined body explicitly that thoseMultiplication is not commutative. In desem case a body is inclined body no longer at the same time. An example is the inclined body of the quaternions, which is not a body.