Body (algebra)

Bodies in connection with selected mathematical sub-areas ( class diagram )

A body is in the mathematical branch of algebra excellent algebraic structure in which the addition , subtraction , multiplication and division can be performed in a certain way.

The term "body" was introduced by Richard Dedekind in the 19th century .

The most important fields used in almost all areas of mathematics are the field of rational numbers , the field of real numbers, and the field of complex numbers . ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$

Formal definition

general definition

A field is a set with two inner two-digit connections " " and " " (which are called addition and multiplication ), for which the following conditions are met: ${\displaystyle K}$ ${\displaystyle +}$ ${\displaystyle \cdot }$

1. ${\displaystyle \left(K,+\right)}$is an Abelian group (neutral element 0).
2. ${\displaystyle {\bigl (}K\setminus \{0\},\cdot {\bigr )}}$is an Abelian group (neutral element 1).
3. Distributive Laws :

   ${\displaystyle a\cdot \left(b+c\right)=a\cdot b+a\cdot c\,}$for everyone . ${\displaystyle a,b,c\in K}$

   ${\displaystyle \left(a+b\right)\cdot c=a\cdot c+b\cdot c\,}$for everyone . ${\displaystyle a,b,c\in K}$

Individual enumeration of the axioms required

A body must therefore fulfill the following individual axioms:

1. Additive properties:
   1. ${\displaystyle a+(b+c)=(a+b)+c}$( Associative law )
   2. ${\displaystyle a+b=b+a}$( Commutative law )
   3. There is an element with ( neutral element ). ${\displaystyle 0\in K}$ ${\displaystyle 0+a=a}$
   4. For each there is the additive inverse with . ${\displaystyle a\in K}$ ${\displaystyle -a}$ ${\displaystyle (-a)+a=0}$
2. Multiplicative properties:
   1. ${\displaystyle a\cdot (b\cdot c)=(a\cdot b)\cdot c}$( Associative law )
   2. ${\displaystyle a\cdot b=b\cdot a}$( Commutative law )
   3. There is an element with ( neutral element ). ${\displaystyle 1\in K\setminus \{0\}}$ ${\displaystyle 1\cdot a=a}$
   4. For each there is the multiplicative inverse with . ${\displaystyle a\in K\setminus \{0\}}$ ${\displaystyle a^{-1}}$ ${\displaystyle a^{-1}\cdot a=1}$
3. Interplay of additive and multiplicative structure:
   1. ${\displaystyle a\cdot (b+c)=a\cdot b+a\cdot c}$(Links- distributive law )
   2. ${\displaystyle (b+c)\cdot a=b\cdot a+c\cdot a}$(Legal distributive law )

Definition as a special ring

A commutative unitary ring that is not the zero ring is a field if every non-zero element in it has an inverse with respect to the multiplication.

In other words, a body is a commutative unitary ring in which the unit group is the same . ${\displaystyle K}$ ${\displaystyle K^{*}}$ ${\displaystyle K\setminus \{0\}}$

Remarks

The definition ensures that addition, subtraction and multiplication work in the "usual" way in a body as well as division with the exception of the non-solvable division by 0 :

• The inverse of in relation to the addition is and is usually called the additive inverse of or the negative of . ${\displaystyle a}$ ${\displaystyle -a}$ ${\displaystyle a}$ ${\displaystyle a}$
• The inverse of in relation to multiplication is and is called the (multiplicative) inverse of or the reciprocal of . ${\displaystyle a}$ ${\displaystyle a^{-1}}$ ${\displaystyle a}$
• ${\displaystyle 0}$is the only element of the body that has no reciprocal value, so it is the multiplicative group of a body . ${\displaystyle K^{\times }=K\setminus \{0\}}$

Note: The formation of the negative of an element has nothing to do with the question of whether the element itself is negative; for example, the negative of the real number is the positive number . In a general body there is no concept of negative or positive elements. (See also ordered body .) ${\displaystyle -2}$ ${\displaystyle 2}$

Generalizations: oblique bodies and coordinate bodies

→ Main articles : oblique bodies  and ternary bodies

If one waives the condition that the multiplication is commutative, one arrives at the structure of the oblique body. However, there are also authors who explicitly assume that the multiplication is not commutative for a skew body. In this case, a body is no longer an oblique body. An example is the oblique body of the quaternions , which is not a body. On the other hand, according to Bourbaki , there are authors who refer to oblique bodies as bodies and the bodies discussed here as commutative bodies.

In analytical geometry , bodies are used to represent the coordinates of points in affine and projective spaces , see Affine coordinates , projective coordinate system . In synthetic geometry , in which spaces (especially planes ) with weaker properties are also examined, generalizations of the oblique bodies, namely alternative bodies , quasi-bodies and ternary bodies , are also used as coordinate areas ("coordinate bodies") .

Properties and terms

• There is exactly one “0” (zero element, neutral element with regard to body addition ) and one “1” (one element, neutral element with regard to body multiplication ) in a body.
• Every body is a ring . The properties of the multiplicative group lift the body out of the rings. If the commutativity of the multiplicative group is not required, one obtains the concept of the oblique body .
• Every field has zero divisors : A product of two elements of the field is 0 if and only if at least one of the factors is 0.
• Each body can be assigned a characteristic that is either 0 or a prime number .
• The smallest subset of a body that still satisfies all body axioms is its prime field . The prime field is either isomorphic to the field of the rational numbers (for fields of the characteristic 0) or a finite residual class field (for fields of the characteristic , especially for all finite fields, see below). ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Z} /p\mathbb {Z} }$ ${\displaystyle p}$
• A body is a one-dimensional vector space about itself as the underlying scalar body. In addition, vector spaces of any dimension exist over all bodies. (→ main article vector space ).
• An important means of examining a field algebraically is the polynomial ring of polynomials in a variable with coefficients . ${\displaystyle K}$ ${\displaystyle K\left[X\right]}$ ${\displaystyle K}$
  • A body is said to be algebraically closed if every non-constant polynomial in can be broken down into linear factors in . ${\displaystyle K}$ ${\displaystyle K\left[X\right]}$ ${\displaystyle K\left[X\right]}$
  • It's called a body completely , if no irreducible non-constant polynomial from multiple in any field extension has zeros. Algebraic closure implies perfection, but not the other way around. ${\displaystyle K}$ ${\displaystyle K\left[X\right]}$
• If a total order is defined in a body that is compatible with addition and multiplication, one speaks of an ordered body and also calls the total order the arrangement of the body. In such bodies one can speak of negative and positive numbers.
  • If in this arrangement every body element can be surpassed by a finite sum of the one element ( ), one says that the body fulfills the Archimedean axiom or that it is Archimedean. ${\displaystyle \alpha }$ ${\displaystyle \alpha <1+1+\cdots +1}$
• In evaluation theory , certain bodies are examined with the help of an evaluation function. They are then called valued bodies.
• As a ring, a body has only the trivial ideals and . ${\displaystyle K}$ ${\displaystyle (0)=\{0\}}$ ${\displaystyle (1)=K}$
• Any non-constant homomorphism from a body to a ring is injective .

Body enlargement

→ Main article : Body enlargement

A subset of a body that forms a body with its operations is called a sub-body. The pair and is called body enlargement , or . For example, the field of rational numbers is a part of the field of real numbers . ${\displaystyle K}$ ${\displaystyle L}$ ${\displaystyle K}$ ${\displaystyle L}$ ${\displaystyle K\subset L}$ ${\displaystyle L/K}$ ${\displaystyle L|K}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {R} }$

A subset of a body is a partial body if it has the following properties: ${\displaystyle U}$ ${\displaystyle K}$

• ${\displaystyle 0_{K}\in U}$, ${\displaystyle 1_{K}\in U}$
• ${\displaystyle a,b\in U\ \Rightarrow \ a+b\in U,\ a\cdot b\in U}$( Closure with regard to addition and multiplication)
• ${\displaystyle a\in U\ \Rightarrow \ -a\in U}$(For every element from there is also the additive inverse in .) ${\displaystyle U}$ ${\displaystyle U}$
• ${\displaystyle a\in U\setminus \{0\}\ \Rightarrow \ a^{-1}\in U}$(For every element from except zero, the multiplicative inverse is also in .) ${\displaystyle U}$ ${\displaystyle U}$

The algebraic subfield that deals with the investigation of body extensions is Galois theory .

Examples

• Well-known examples of bodies are
  • the field of rational numbers , ie the set of rational numbers with the usual addition and multiplication ${\displaystyle (\mathbb {Q} ,+,\cdot )}$
  • the field of real numbers , ie the set of real numbers with the usual addition and multiplication, and ${\displaystyle (\mathbb {R} ,+,\cdot )}$
  • the field of complex numbers ie the set of complex numbers with the usual addition and multiplication. ${\displaystyle (\mathbb {C} ,+,\cdot )}$

• Bodies can be expanded through adjunction . An important special case - especially in Galois theory - are algebraic body extensions of the body . The expansion body can as a vector space over to be construed. ${\displaystyle \textstyle \mathbb {Q} }$ ${\displaystyle \textstyle \mathbb {Q} }$
  • ${\displaystyle \textstyle \mathbb {Q} ({\sqrt {2}})=\{a+b{\sqrt {2}}\mid a,b\in \mathbb {Q} \}}$is a body. It suffices to show that the inverse of is also of the given form: One possible basis of is { }. ${\displaystyle \textstyle a+b{\sqrt {2}}\neq 0}$
          ${\displaystyle {\frac {1}{a+b{\sqrt {2}}}}={\frac {(a-b{\sqrt {2}})}{(a+b{\sqrt {2}})\cdot (a-b{\sqrt {2}})}}={\frac {(a-b{\sqrt {2}})}{(a^{2}-2b^{2})}}={\frac {a}{(a^{2}-2b^{2})}}+{\frac {-b}{(a^{2}-2b^{2})}}{\sqrt {2}}}$
    ${\displaystyle \textstyle \mathbb {Q} ({\sqrt {2}})}$ ${\displaystyle \textstyle 1,{\sqrt {2}}}$
  • ${\displaystyle \mathbb {Q} ({\sqrt {2}},{\sqrt {3}})=\{a+b{\sqrt {2}}+c{\sqrt {3}}+d{\sqrt {6}}\mid a,b,c,d\in \mathbb {Q} \}}$is a body with a base { }. ${\displaystyle \textstyle 1,{\sqrt {2}},{\sqrt {3}},{\sqrt {6}}}$

• The remainder class fields with prime numbers and provide further examples ${\displaystyle \mathbb {Z} /p\mathbb {Z} =\mathbb {F} _{p}}$ ${\displaystyle p}$
  • their finite body extensions , the finite bodies ,
  • more generally their algebraic field extensions, the Frobenius fields , and
  • more generally their arbitrary field extensions, the fields with prime number characteristics .
• For every prime number the fields of the p-adic numbers . ${\displaystyle p}$ ${\displaystyle \mathbb {Q} _{p}}$
• The set of whole numbers with the usual connections is not a field: it is a group with a neutral element and each has the additive inverse , but is not a group. After all, the neutral element is, but apart from and there are no multiplicative inverses (for example is not a whole, but a really rational number): ${\displaystyle (\mathbb {Z} ,+,\cdot )}$ ${\displaystyle (\mathbb {Z} ,+)}$ ${\displaystyle 0}$ ${\displaystyle a\in \mathbb {Z} }$ ${\displaystyle -a}$ ${\displaystyle (\mathbb {Z} \setminus \{0\},\cdot )}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle -1}$ ${\displaystyle 3^{-1}=1/3}$
  • The whole numbers simply form an integrity ring , the quotient fields of which are the rational numbers.
• The concept with which the integrity ring of whole numbers can be expanded to form the field of rational numbers and embedded in it can be generalized to any integrity ring:
  • In function theory , the integrity ring of the holomorphic functions in one area of the complex number level gives rise to the solids , the meromorphic and abstract functions in the same area
  • from the integrity ring of the formal power series over a body its quotient field, analogously from the integrity ring of the formal Dirichlet series ${\displaystyle K\left[[x\right]]}$ ${\displaystyle K}$
  • from the ring of polynomials in variables, whose quotient field, the field of rational functions in as many variables. ${\displaystyle n}$ ${\displaystyle K\left[x_{1},x_{2},\dots ,x_{n}\right]}$ ${\displaystyle K\left(x_{1},x_{2},\dots ,x_{n}\right)}$

Finite bodies

→ Main article : Finite body

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A body is a finite body if its basic set is finite. The finite fields are fully classified in the following sense: Every finite field has exactly elements with a prime number and a positive natural number . With the exception of isomorphism, there is exactly one finite body for each such body, which is denoted by. Every body has the characteristic . The addition and multiplication tables of the are shown here as an example ; the lower part of the body highlighted in color . ${\displaystyle K}$ ${\displaystyle q=p^{n}}$ ${\displaystyle p}$ ${\displaystyle n}$ ${\displaystyle q}$ ${\displaystyle \mathbb {F} _{q}}$ ${\displaystyle \mathbb {F} _{p^{n}}}$ ${\displaystyle p}$ ${\displaystyle \mathbb {F} _{4}}$ ${\displaystyle \mathbb {F} _{2}}$

In the special case we get for every prime number the field that is isomorphic to the remainder class field . ${\displaystyle n=1}$ ${\displaystyle p}$ ${\displaystyle \mathbb {F} _{p}}$ ${\displaystyle \mathbb {Z} /p\mathbb {Z} }$

history

Essential results of body theory are due to Évariste Galois and Ernst Steinitz .

See also

• Algebraic number field

literature

• Siegfried Bosch : Algebra. 7th edition. Springer-Verlag, 2009, ISBN 3-540-40388-4 , doi: 10.1007 / 978-3-540-92812-6 .
• Thomas W. Hungerford: Algebra. 5th edition. Springer-Verlag, 1989, ISBN 0-387-90518-9 .

Web links

Wikibooks: Math for Non-Freaks: Body Axioms  - Learning and Teaching Materials

Wikibooks: Math for non-freaks: Inferences from the body axioms  - learning and teaching materials

Wiktionary: body  - explanations of meanings, word origins, synonyms, translations

Individual evidence

1. ↑ Any solution to every equation violates the ring axioms. ${\displaystyle x}$ ${\displaystyle 0\cdot x=a\in K}$
2. ^ Albrecht Beutelspacher : Lineare Algebra . 7th edition. Vieweg + Teubner Verlag , Wiesbaden 2010, ISBN 978-3-528-66508-1 , p. 35-37 .