Division ring

A skew fields and division ring is an algebraic structure having all the features of the body, except that the multiplication is not necessarily commutative.

A skew field is thus a ring with identity in which every element has a multiplicative inverse.

For every skew field (as for every other ring also ) defines a characteristic.

All skew field with a finite number of elements are, by the theorem of Wedderburn at the same time body. If a skew field no body, it must therefore consist of infinitely many elements. One example is the skew field of quaternions. It has the characteristic 0

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

There are non-commutative division ring, which allow the shortcuts of skewfield acceptable, total arrangement. They are called arranged skew field.

For the algebraic description of an affine plane or a projective plane skew field are used as coordinate ranges in synthetic geometry for desarguesche levels. Among other things, an alternative body, body and quasi Ternärkörper be there to describe nichtdesarguescher ( affine or projective ) planes for same purpose. The term skew field is generalized: Every division ring is an alternative body of each alternative body a quasi body and each body a quasi Ternärkörper.

• 2.1 Partial Body
• 2.2 center and centralizer
• 2.3 characteristics
• 2.4 Morphisms and ideals 2.4.1 Antihomomorphismen

• 3.1 Equivalent description by a positive field
• 3.2 Arrangement ability

• 4.1 An example class according to Hilbert
• 4.2 Non-commutative skew field of arbitrary characteristic
• 4.3 Two concrete noncommutative skew field 4.3.1 A skew field of characteristic 2
• 4.3.2 A skew field of characteristic 0
• 4.3.3 uncountability the two sample chief body

History of the term

As the first noncommutative body Quaternionenring 1843 by Sir William Hamilton was constructed. His aim was to represent vectors of the three-dimensional space and, if possible, analogous to the representation of vectors of the plane by complex numbers. Hamilton and his successors built on this basis to a sophisticated geometric calculus, which ultimately led to the development of vector analysis. Skew field such as the quaternions, the finite-dimensional vector spaces are over their center, were investigated extensively in the 1920s and 1930s, the area was revived in the 1970s.

The first division ring, which is infinite-dimensional over its center, was designed by David Hilbert in 1903. His concern was to be able to define a model for a non-commutative division ring that allows an arrangement which is analogous compatible to the known arrangements of the formally real ( commutative ) body with the algebraic links. Then such a skew field he could define an affine geometry that meets some, but not all the axioms of his axioms of Euclidean geometry.

1931 Øystein Ore studied the below described in this article and named after him design method for skew fields.

Language rules

In the older literature also non-commutative division ring are referred to as "body" often, the term " skew field " was only used when it should be emphasized that a certain "body" (division ring ) is not commutative. In French, the term " corps" to this day the non- commutative case includes with.

Definitions and characteristics

A set with two binary operations (addition), (multiplication ) and two constants is called skew field if the following axioms hold:

Equivalent to this axiom system is the following, which does not require distributive law:

There are provided as above and

Then is a skew field.

Equivalently, is this definition:

A ring is called skew field if

Here, it is not required that the equations possess unique solutions, the uniqueness can be shown, however. A skew field is thus a ring in which a left and a right division can be defined, hence the name division ring.

The now following equivalent axiom system emphasizes the multiplicative aspect of the skew field:

It is a group. The group with 0 on G is the set with the link continued by the agreement. Is now an image with

Then with the addition

A skew field. Given a skew field with addition of the mapping is given by.

Part of body

If S is a skew field and a subset with and is a subset of, as well as a subset of, then we say that T is a subfield of S. For these sub- body relationship to write then

Center and centralizer

• Is a division ring, then that means the amount the center of S.
• Elements are referred to as central elements of the skew field.
• The center of S is the center in terms of the group theory of the multiplicative group together with the zero element.
• The centralizer of a subset defined by each centralizer is a (not necessarily commutative ) subfield of S.
• For the centralizer of a subset A is always
• The centralizer versa subset relations to: . Especially applies.

Characteristics

The characteristic of a swash body is defined analogously to the bodies of commutative:

• If it means the smallest positive integer with this property characteristic of. This must then be a positive prime number.
• Is, then one defines, has the characteristic 0

Morphisms and ideals

The concept of homomorphism is for skew fields defined so precisely as the term homomorphism in the ring theory: If a division ring and a ring, then is called a homomorphism if for all:

About the general properties of a ring homomorphism addition has the following properties, since a skew field is:

A ring homomorphism is called Schiefkörperhomomorphismus, if a skew field is as Schiefkörperisomorphismus if it is bijective and as Schiefkörperautomorphismus if, in addition still is.

Antihomomorphismen

Is a noncommutative, ie " real" skew field, then the Antihomomomorphismen of interest in addition to the ring homomorphisms: Is again a skew field and a ring, it means Anti ( ring ) homomorphism if for all:

For commutative body, the course is not different from the notion of homomorphism, because the commutative property of multiplication is transferred to the image.

All of these terms for homomorphisms be made according to Antihomomorphismen, the trivial " anti - " homomorphism coincides with the trivial homomorphism match. There is no need of Antiautomorphismus exist ( or be known ) in general. For the real Quatornionenschiefkörper conjugation is a Antiautomorphismus, as are the analogous defined mapping for quaternionenartigen skew fields which are mentioned in the examples in this article. For every skew field but you can antiisomorphe a structure, construct its mating ring by reversing the multiplication, so you defined for and retain the original in addition. Then a antiisomorpher to skew field, the mediating Antiisomorphismus is the identity map on the set.

Properties and related terms

• A division algebra multiplication need not necessarily be associative. Every division ring is a division algebra over its center, a K- division algebra over a field K is a skew field if and only if the associative law is fulfilled and thus forms a group. In this case, K is a subfield of the center of D,
• Every division ring is a fast body, a body is almost exactly then a skew field if it satisfies both distributive laws.
• If not called for in the axiom system of Cohn with the successor to the Figure 3 axiom, then it describes a fast body.
• Every division ring is a half body in terms of geometry and an alternative body, half body or alternative body is exactly then a skew field if the multiplication is associative.
• A ring with identity ( unitary ring ) is a skew field if and only if every element except the zero element has a left and a right inverse element with respect to multiplication. The equality of these two inverse elements, and the uniqueness of the words at the same time left and right inverse element can then be proved from the other ring axioms.

In the Assigned skew field

A skew field on which a total order is defined is called arranged skew field if the order with the field operations is acceptable. Compatibility here means that the following axioms of order apply to all:

• Follows ( monotony of the addition) and
• And from this and ( seclusion of the positive range with respect to multiplication ).

This means, as usual, that is. It is the weak total ordering associated strict total ordering.

The additive group is a commutative group arranged in an arranged skew field and therefore must be torsion-free. Therefore, the characteristic of an arranged skew field is always 0 but this is not a sufficient condition for the arrangement ability to compare also the article Parent body. The Quaternionenschiefkörper does not allow any arrangement!

Equivalent description by a positive field

Is an ordered skew field and its strict, total order relation, then we define:

One then writes

It follows from the Trichotomiegesetz that each number is in exactly one of the two sets, because you can compare each number such as 0. For compatibility with the addition follows:

For compatibility with the addition and transitivity follows for:

For compatibility with the multiplication follows immediately.

The three properties of the positive range characterize the arrangement on the skew field completely. It is namely:

A skew field can be precisely then to an array if it contains a subset with the following three properties:

An arrangement of, namely the arrangement with the positive range is then added to the definition of the partial order. A proof of this theorem, in which only provided by the structure is that it is a ring with unit element, is found in the textbook by Fuchs.

Arrangement ability

The characterization of the arrangement through a positive area is often useful to construct an arrangement on a given skew fields and even more suitable to prove that a given division ring does not allow configuration. For this purpose, some of the properties of the positive region, ie, a subset of the characteristics 1 to 3 of a positive range, useful:

• From the first property follows, since it is said there union is therefore always disjoint.
• For arbitrary, because one of the elements is. Quantity theory formulated. Has an ordered skew fields the property that every positive element is a square number, then there is only accurate to this an array. → This property characterized ( under the commutative skew fields ) the Euclidean body.

• Is an ordered subfield of, is for the a perfect square and is (in terms of order on ) is negative, then there is at least no arrangement on that. Arrangement to continue Allows only one arrangement, then you can not be placed under these conditions. Thus, for example, the above statement that the arrangement does not allow Quaternionenschiefkörper be proved: that the real number may be as Euclidean body, only one arrangement to and there are ( infinitely many ) elements.

• If, then we also have, for otherwise and contrary to.
• Together with the seclusion (3rd property) it follows that a subgroup of the multiplicative group.
• Since, according to the first property is the only real left and right coset of, is a normal subgroup of index 2 in the multiplicative group.

Construction and examples

Commutative body can be produced from given bodies by algebraic or transcendental field extensions, each such body goes from the prime field of its characteristics produced by a combination of these two kinds of extension. A similar " canonical " method to construct non-commutative division ring is not known. Most methods are based on a ( suitable ) noncommutative embedding zero-divisor -free ring in his right or left quotient skew field. A relatively simple sufficient criterion to a ring found Øystein Ore with the eponymous Ore condition.

An example class according to Hilbert

Infinite-dimensional extensions can be constructed analogously to the skew field specified by Hilbert. This looks like this:

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

A generalization of Hilbert's construction used in place of other Ringendomorphismen of.

Non-commutative skew field of arbitrary characteristic

A variant of Hilbert's idea comes out with a one step extension of a body, if it allows a non-identical Körperautomorphismus. These include, for example, all finite body, (see Frobeniushomomorphismus ), all real Galois extension field of the rational number field, especially the quadratic extension field.

In the design to go from the formal Laurent series over finite principal part of, so the formal functions:

Addition is defined by the rows of usual component- wise addition of the coefficients. The product is for by

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

One lists the structure of the set of formal Laurent series with ordinary addition and multiplication as the modified and calls him english skew Laurent series ring in one indeterminate. ( No known German name. ) This ring is (if the defining Körperautomorphismus is not the same as ) a noncommutative division ring with the same characteristics as the original body.

Two concrete noncommutative skew field

A skew field of characteristic 2

The smallest output body coming for the described "skew Laurent series ring" structure into consideration, the body with four elements. You can win from him by the adjoint one root in the irreducible polynomial. Then is not the identity element and thus, as 3 is a prime number, a generating element of the three-element cyclic multiplicative group. The only non-identical automorphism of the multiplicative group is uniquely determined by the last equation follows from the fact that zero of is. This Gruppenautomorphismus continues by agreeing to a non-identical Körperautomorphismus of and is a concrete example of a noncommutative skew field of characteristic 2

A skew field of characteristic 0

Here you have to expand the field of rational numbers at least once a square. We choose. Then is given by a non-identical Körperautomorphismus of. This is a noncommutative skew field of characteristic 0

In addition one can note that the commutative initial body (perhaps contrary to the intuitive notion of a body part ) allows two different arrangements, on the other hand as a prime field only. You have to decide whether the adjoint " square root " the positive or negative zero to be of rational polynomial. We first decide. Where exactly then is the arrangement of, is then defined as the operation is on a arranged skew field (due to the properties of the positive area shown above) is strictly increasing for positive elements of the range, so for example, must apply for, etc.

Man charged with two simple squares with the above given product definition:

Now would both elements as different from 0 - square numbers in the positive range of lie, as well as the number, firstly because this is also a square number in, and secondly, because the rational numbers allow only one arrangement. This results in a conflict with the above-mentioned sub-group characteristics of a positive range.

These considerations are apparently quite regardless of which selects the two possible arrangements for you.

Uncountability the two sample chief body

Both skew fields each contain as subsets of the uncountable sets

Whose coefficient sequences consist only of the "numbers" 0 and 1 and can therefore be interpreted as binary representations of all real numbers. Both of them are so according to Cantor's second diagonal argument uncountable subsets of their skew fields that are themselves therefore also uncountable sets.

One sees now easy to see that this argument is constructed skew field applies to each according to the described "skew Laurent series ring" method.

Quaternionenartige skew field

One can use any commutative field to carry out the construction of the Hamiltonian skew field of real quaternions general instead of whose characteristic is not 2. ( The " signs " are important for the design. ) For formally real body, this results in a real division ring. As can be seen from the detailed information and references in the article quaternion is obtained by the construction of a structure which always has the following characteristics:

With these three design steps so you always get a four-dimensional algebra. That each element of the multiplication in the interior of elements of commutation, also results from the design.

The standard function

Assumes only values ​​of the base body.

For an inverse formation in must now be divided by such standard values ​​. The coefficients can be arbitrary elements of ( except that all can not be 0, because the zero element has and will continue to need no inverse ). Therefore, inverse as elements exist if and only if it is not represented in the zero element as nichtriviale sum of ( here at most 4 ) square numbers. It is then

This is a non-commutative division ring when a formal real body. This is four-dimensional skew field over its center. He does not allow any arrangement, because of the elements in what makes the existence of a positive area impossible.

If you choose the base a countable body, for example, then you therefore also has a countable real skew field.

If (as a vector space over ) finite-dimensional formally real extension field, that is true and then all nontrivial endomorphisms of bijective, so Schiefkörperautomorphismen and at the same time Vektorraumautomorphismen of. They can therefore, represent the choice of a fixed - base of by regular matrices. Thus, this group of Schiefkörperautomorphismen is represented as a subset of, the general linear group, because it is then.

Impossible is the invertibility for all elements, however, over fields of characteristic. It is sufficient to show that such elements with coefficients in the prime field exist whose standard value is 0. For that is given with. Now let therefore an odd prime number. We must show then that the congruence has a nontrivial solution. This is relatively easy to prove by counting, for example, by this argument drawer.