Oblique body

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A skew field, or division ring, is an algebraic structure that has all of the properties of a field except that multiplication is not necessarily commutative.

An oblique body is thus a ring with a single element , in which each element has a multiplicative inverse .

As such, the characteristic is defined for him .

Every oblique field with a finite number of elements is already a field according to Wedderburn's theorem , that is, the multiplication is automatically commutative. If a sloping body is not a body, it must contain an infinite number of elements. An example is the oblique body of the quaternions , it has the characteristic 0.

The center of an inclined body is a (commutative) field , and by means of inclusion it becomes an - algebra . The sum of all skew field with a predetermined center , as - vector space are finite, is by the Brauer group of described.

There are non-commutative oblique bodies that allow a total arrangement that is compatible with the links of the oblique body . They are referred to as arranged oblique bodies .

For the algebraic description of an affine plane or a projective plane , inclined bodies are used as coordinate areas in synthetic geometry for Desargue planes . To describe non-desargue (affine or projective) planes, alternative bodies , quasi-bodies and ternary bodies , among others , are used there for the same purpose . The term oblique body is generalized: every oblique body is an alternative body, every alternative body is a quasi-body and every quasi-body is a ternary body.

History of the term

As the first non-commutative body, the quaternion ring was constructed by Sir William Rowan Hamilton in 1843 . His aim was to represent the three-dimensional space vectors specifically as possible to the representation of vectors of the plane by complex numbers . Hamilton and his successors built a sophisticated geometric calculus on this basis, which ultimately led to the development of vector analysis . Oblique bodies like the quaternions, which are finite-dimensional vector spaces above their center , were extensively researched in the 1920s and 1930s, and the area was revived in the 1970s.

The first oblique body, which is infinitely dimensional above its center, was constructed by David Hilbert in 1903. His aim was to be able to specify a model for a non-commutative inclined body that would allow an arrangement that is compatible with the algebraic connections analogous to the known arrangements of formally real (commutative) bodies . Using such an oblique body, he could then define an affine geometry that fulfills some, but not all, of the axioms of his axioms of Euclidean geometry .

In 1931 Øystein Ore studied the construction method for oblique bodies, which is described later in this article and named after him.

Language regulations

In the older literature, non-commutative oblique bodies are also often referred to as "bodies": In this usage, a body in today's sense was called a "commutative body", a oblique body in today's sense simply as a "body" (or "not necessarily commutative body") ) and only the real oblique body or non-commutative oblique body as "oblique body"; Wedderburn's sentence usually means “Every finite oblique body is a body”, but in this usage “Every finite body is commutative” or “There are no finite oblique bodies”. In French, the term “corps” includes the non-commutative case to this day, as does the term “corpo” in Italian, the latter parallel to the (the English expression for a body, field , corresponding) term “campo” for body in the sense of this article is used.

Definitions and characteristics

A set with two two- digit operations (addition), (multiplication) and two constants is called an oblique field if the following axioms hold:

  1. is a commutative / Abelian group .
  2. is a group .
  3. The two distributive laws apply
and for everyone

Equivalent to this system of axioms is the following, which does without a distributive law:

Let it be assumed as above as well

  1. an abelian group,
  2. a group,
  3. with the star product given by a group and
  4. it applies

then is a sloping body.

This definition is also equivalent to this:

A ring is called an oblique body if

  1. the equations
are forever solvable in

It is not required here that the equations have unique solutions, but the uniqueness can be shown. An inclined body is a ring in which a left and a right division can be defined, hence the name division ring .

The following, equivalent axiom system emphasizes the multiplicative aspect of the oblique body:

It is a group. The group with 0 on is then the set with the link continued by the agreement . Is now a figure with

  1. For
  2. For

then with the addition

an oblique body. For a given inclined body with addition, the mapping is given by.

Partial body

If a skew body and a subset with and is a subgroup of and a subgroup of , then one calls a subfield of . You then write for this partial body relationship

Center and centralizer

  • If the body is inclined, then the set is called the center of .
  • Elements are called central elements of the oblique body.
  • The center of the center in terms of group theory the multiplicative group together with the zero element: .
  • The centralizer of a subset is defined by Each centralizer is a (not necessarily commutative) subfield of .
  • The following always applies to the centralizer of a subset
  • The centralizer versa subset relations to: . Especially applies .


The characteristic of an inclined body is defined analogously to that of commutative bodies:

  • The smallest positive natural number with the property is called characteristic of . This must then be a positive prime number .
  • If for all positive natural numbers one defines: has the characteristic 0.

Morphisms and ideals

The term homomorphism for oblique bodies is defined in exactly the same way as the term ring homomorphism in ring theory : If there is a oblique body and a ring, then it is called a ring homomorphism if the following applies to all :

and .

In addition to the general properties of a ring homomorphism , since it is a skew body it has the following properties :

  1. It's either the zero ring or is injective , so one embedded in the ring , as has no other than the trivial ideals , .
  2. In the case of embedding the ring is carried naturally to a - Links module , a basis and a unique dimension than does so to a free module over
  3. If surjective and not the zero ring, then is isomorphic to and itself a skew.
  4. Is , then one calls an oblique body endomorphism, even if is. But if the endomorphism is injective, then in general it does not need to be surjective. If a partial body is fixed by point and is finite, then bijectivity follows from surjectivity .

A ring homomorphism is referred to as a sloping body homomorphism if it is also a sloping body, a sloping body isomorphism if it is bijective and a sloping body automorphism if it is beyond .


If a non- commutative, i.e. "real", inclined body, then, in addition to the ring homomorphisms, the antihomomorphisms are of interest: If there is again an inclined body and a ring, then anti (-ring) homomorphism is called if the following applies to all :

and .

For commutative bodies, of course, this does not differ from the concept of ring homomorphism, because the commutative law of multiplication is transferred to the image .

All named terms for homomorphisms are formed accordingly for antihomomorphisms, the trivial "anti" homomorphism corresponds to the trivial homomorphism. In general, no antiautomorphism of has to exist (or be known). For the real quatornion oblique body, the conjugation is an antiautomorphism, as is the analogously defined mapping for the quaternion-like oblique bodies, which are mentioned in the examples in this article . For every oblique body one can construct an anti-isomorphic structure, its counter-ring , by reversing the multiplication, so one defines for and keeps the original addition. Then there is a misaligned body that is too anti-isomorphic , the mediating anti-isomorphism is the identical mapping on the set .

Properties and related terms

  • In a division algebra, the multiplication need not necessarily be associative. Every oblique body is a division algebra over its center, a division algebra over a body is a oblique body if and only if the associative law is fulfilled and thus forms a group. In this case a partial body of the center of ,
  • Every oblique body is an almost body , an almost body is an oblique body if and only if it fulfills both distributive laws.
  • If the third axiom is not required in Cohn's system of axioms with the successor mapping, then it describes an almost body.
  • Every oblique body is a half body in the sense of geometry and an alternative body, a half body or alternative body is a oblique body if and only if the multiplication is associative.
  • A ring with a one element ( unitary ring ) is an inclined body if and only if every element apart from the zero element has a left and a right inverse element with respect to the multiplication. The equality of these two inverse elements and the uniqueness of the left and right inverse element can then be proven from the other ring axioms.

Arranged oblique body

An oblique body on which a total order is defined is called an arranged oblique body if the order is compatible with the body operations. Compatibility here means that the following axioms of arrangement apply to all of them :

  • follows from (monotony of addition) and
  • from and follows and (isolation of the positive region with respect to multiplication).

The requirement that the order should be a "total order" means:

  1. The two-digit relation on is reflexive , that is, it applies to every element and
  2. it is transitive , that is, it always follows from . With these two properties, the relation is a weak partial order on the set . It should now also be total , that means:
  3. Any oblique body elements are always comparable in terms of size, so it must apply to any :
and . The requirement is equivalent
It always applies to exactly one of the three relations . This is the so-called trichotomy law.

As usual, this means that is. It is the strict total order assigned to the weak total order.

The additive group is a commutative, arranged group in an arranged inclined body and must therefore be torsion-free . Therefore, the characteristic of an arranged oblique body is always 0. This is not a sufficient condition for the ability to be arranged, see also the article Ordered body . The quaternion oblique body does not allow any arrangement!

Equivalent description with a positive area

If there is an arranged oblique body and its strict, total order relation, then one defines:

and names the positive range of , an element of is then called positive, positive element of or a positive number.

You then also write

and names the elements of negative etc.

From the law of trichotomy it follows that every number lies in exactly one of the two sets , because every such number can be compared with 0. From the compatibility with the addition follows:

, as it corresponds to the intuitive notion of "negative numbers". So one has and this union is even a disjoint union .

From the compatibility with addition and transitivity it follows for :

, that is .

From the compatibility with the multiplication it follows immediately .

The three properties of the positive area completely characterize the arrangement on the inclined body. The following applies:

A sloping body permits an arrangement if and only if it contains a subset with the following three properties:

  1. and ,
  2. ,
  3. .

An arrangement of , namely the arrangement having the positive area is then given by the definition of the partial order on given. A proof of this theorem, in which the structure is only assumed to be a ring with a one element, can be found in Fuchs' textbook.


The characterization of the arrangement by a positive area is often suitable for constructing an arrangement on a given inclined body and even better suitable for proving that a given inclined body does not permit an arrangement. For this purpose, some properties of the positive area , i.e. a subset of the properties 1st to 3rd of a positive area, are useful:

  • From the 1st property it follows , because it is , the union named there is therefore always disjoint.
  • For any is , because one of the elements is in . Quantity theory formulated: . If an arranged oblique body has the property that every positive element is a square number, then only this one arrangement exists . → This property characterizes (among the commutative oblique bodies) the Euclidean bodies .
  • If an arranged partial body is of , is the square number for one and is negative (with respect to the order on ) , then in any case no arrangement on exists which continues the arrangement on . If only one order is allowed, then under these conditions no order can be made. This allows, for example, the above statement that the Quaternionenschiefkörper not allow any arrangement, be proved: that the real number can be as Euclidean body, only an arrangement to and there are (infinitely many) elements with .
  • Is , then also applies , because otherwise and would be contrary to .
  • Together with the closure (3rd property) it results that a subgroup is the multiplicative group .
  • Since, according to the 1st property, is the only real left and right subsidiary class of , there is a normal divisor of index 2 in the multiplicative group.

Construction and examples

Commutative bodies can be generated from given bodies through algebraic or transcendent body extensions ; each such body emerges from the prime field of its characteristic through a combination of these two types of extensions. A comparable "canonical" method of constructing non-commutative oblique bodies is not known. Most methods are based on embedding a (suitable) non-commutative, zero-divisor-free ring in its right or left quotient oblique. Øystein Ore found a relatively simple, adequate criterion for a ring with the Ore condition named after him .

An example class according to Hilbert

Infinite-dimensional extensions can be built up analogously to the inclined body given by Hilbert. This looks like this:

  1. Be a crooked body or body
  2. the rational body of functions in a central indeterminate .
  3. On the mapping defined by is a ring endomorphism.
  4. From this the non-commutative polynomial ring is formed with a new indeterminate , on which the multiplication of with is determined by the commutation relation ( interchanged with elements of the initial field ).
  5. is the right quotient oblique body of the zero divisor-free Ore-ring and is called the Hilbert body.

The center is also the center of the Hilbert body and it is always . If it is a formally real (commutative) body, then an arrangement compatible with the algebraic links allows .

A generalization of Hilbert's construction uses ring endomorphisms of instead of other .

Non-commutative oblique bodies of any characteristic

A variant of Hilbert's idea manages with a one-step expansion of a body , provided that it allows a non-identical body automorphism . This includes, for example, all finite fields , where is (see Frobenius homomorphism ), all true Galois extension fields of the rational number field , especially the square extension fields .

In design work goes from the formal Laurent series over finite main part of, so the formal functions:

The addition is defined by the component-wise addition of the coefficients that is usual for series. The product is for by

Are defined.

(For is the -fold application of the inverse automorphism, is the identical automorphism of .)

The structure from the set of these formal Laurent series with ordinary addition and the modified multiplication is noted as and called skew Laurent series ring in one indeterminate in English . (No German name known.) This ring is (if the defining body automorphism is not identical) a non-commutative oblique body with the same characteristics as the starting body .

Two concrete non-commutative oblique bodies

An oblique body of characteristic 2

The smallest starting body that comes into consideration for the described “skew Laurent series ring” construction is the body with four elements. You can him win by a zero of in irreducible polynomial adjoint : . Then it is not the one element and thus, since 3 is a prime number, a generating element of the three-element cyclic multiplicative group . The only non-identical automorphism of this multiplicative group is uniquely determined by, the last equation results from the fact that is zero . This group automorphism is continued by the agreement to a non-identical body automorphism of and is a concrete example of a non-commutative oblique body of characteristic 2.

A sloping body with characteristic 0

Here you have to expand the field of rational numbers at least once by the square . We choose . Then is given by a non-identical body automorphism of . This means that there is a non-commutative inclined body with characteristic 0.

The inclined body does not allow any arrangement.

In addition , one can note that the commutative initial body (perhaps contrary to the intuitive idea of ​​a partial body ) allows two different arrangements , whereas as a prime body only one. You have to decide whether the adjoint “square root” should be the positive or negative zero of the rational polynomial . We decide first . Where exactly then lies in the arrangement of is then determined, because the function is strictly monotonically increasing for elements of the positive area on an arranged oblique body (due to the properties of the positive area shown above) , so, for example , must apply because of , etc.

One calculates with two simple square numbers with the product definition given above:

Now both elements, as square numbers different from 0, should be in the positive range of, but also the number , firstly because it is also a square number in and secondly because the rational numbers only allow one arrangement. This leads to a contradiction to the above-mentioned subgroup properties of a positive area.

These considerations are evidently quite independent of which of the two possible arrangements one chooses.

Countability of the two example oblique bodies

Both inclined bodies contain the uncountable sets as subsets


whose coefficient sequences consist only of the "numbers" 0 and 1 and can therefore be interpreted as binary representations of all real numbers . According to Cantor's second diagonal argument, both of them are uncountable subsets of their oblique fields, which are therefore also uncountable sets themselves.

It is now easy to see that this argument applies to every oblique body constructed according to the “skew Laurent series ring” method described.

Quaternion-like oblique bodies

The construction of the Hamiltonian inclined field of the real quaternions can be carried out more generally with an arbitrary commutative field in place of whose characteristic is not 2. (The “signs” are important for the construction.) For formally real bodies this results in a real oblique body. As you can see from the detailed information and references in the Quaternion article , the construction gives you a structure that always has the following properties:

  1. The multiplication with elements from makes a four-dimensional vector space, in particular the multiplication with elements from fulfills both distributive laws . This is how the construction is set up: The symbols are introduced as formal identifiers for four basis vectors.
  2. The “inner multiplication” in is defined by the Hamiltonian relations and for basis vectors and then continued distributively to any elements . This inner multiplication thus also fulfills both distributive laws according to construction.
  3. The internal multiplication of scalar multiples of the basis vectors still fulfills the associative law, because the elements with the Hamiltonian relations and the interpretations of the formal signs ( initially with regard to the group ) by the additional relations form a group, the quaternion group. Since this group is not commutative, the internal multiplication does not satisfy the commutative law either.

With these three design steps can therefore always receives a four-dimensional - algebra . The fact that every element of commutates with elements of in the internal multiplication also results from the construction.

The norm function

only accepts values ​​from the basic body.

For an inverse formation in, it must now be possible to divide by such standard values . The coefficients can be any elements (except that not all can be 0, because the zero element does not have and does not need an inverse). Therefore inverses exist for any elements if and only if the zero element cannot be represented as a nontrivial sum of (here at most 4) square numbers. It is then

with .

This becomes a non-commutative inclined body if is a formally real body . This oblique body is four-dimensional above its center . It does not allow any arrangement, because for the elements is what makes the existence of a positive area impossible.

If one chooses a countable body as the base body , for example then one also has a countable real oblique body .

If a finite-dimensional, formally real extension field (as a vector space over ) is, that is, if and , then all nontrivial endomorphisms of bijective, i.e. oblique body automorphisms and at the same time vector space automorphisms of . So they can be represented by regular matrices after choosing a fixed basis of . Thus the group of these oblique body automorphisms is represented as a subgroup of , the general linear group , because it is then .

In contrast, it is impossible for all elements to be inverted over bodies of a characteristic . To do this, it suffices to show that there are elements with coefficients from the prime field whose norm value is 0. For that is given. Now let so an odd prime number . It then has to be shown that the congruence has a nontrivial solution. This can be proven relatively easily by counting, for example using this drawer argument .


To the order-theoretical definitions and statements
  • László Fuchs: Partly ordered algebraic structures . Vandenhoeck u. Ruprecht, Göttingen 1966, ISBN 3-540-56801-8 .
  • Sibylla Prieß-Crampe : Arranged structures . Groups, bodies, projective levels (=  results of mathematics and its border areas . Volume 98 ). Springer, Berlin / Heidelberg / New York 1983, ISBN 3-540-11646-X .

References and comments

  1. The attribute “non-commutative” always refers to the multiplication of oblique bodies . Non-commutative oblique bodies are often called “real oblique bodies” because they are not bodies .
  2. ^ Jacobson (1996).
  3. See for example the remarks on this in van der Waerden: Algebra I. (1993), 1. § 11 rings.
  4. G. Pickert in: Mathematische Zeitschrift. No. 71, 1959, pp. 99-108.
  5. van der Waerden: . Algebra II § 97, p 57. Some authors write for also
  6. van der Waerden: Algebra I. § 11, p. 38 f.
  7. ^ Cohn (1995).
  8. Systematically the term “part skewed body” would be better here , but this is hardly used in the literature, cf. Pickert (1951).
  9. Note that a sum with summands is meant. This has to be distinguished from the multiplication of two oblique body elements!
  10. Prieß-Crampe (1983).
  11. Prieß-Crampe (1983), II § 1 sentence 1. She even formulates this sentence there for ring with one element, the multiplication of which does not necessarily have to be associative.
  12. Fuchs (1966), p. 163.
  13. a b Cohn (1995), 6.1.
  14. ^ A b c Lars Kadison and Matthias T. Kromann: Projective Geometry and Modern Algebra . Birkhäuser, Boston / Basel / Berlin 1996, ISBN 3-7643-3900-4 , 7.3: A Division Ring with Characteristic p ( table of contents PDF [accessed on August 6, 2013]).
  15. You have to formulate a bit laborious here, because the elements 0, 1 are off in the first case and rational numbers in the second.
  16. More precisely: The non-terminating coefficient sequences are sufficient for this .
  17. From these additional relations it follows with the Hamilton relations that the two elements already create the ring .