Linked field

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In mathematics, a linked field is a field for which the quadratic forms attached to quaternion algebras have a common property.

In mathematics, a quadratic form is a polynomial with terms all of degree two. For example,

In mathematics, a quaternion algebra over a field F is a central simple algebra A over F that has dimension 4 over F. Every quaternion algebra becomes the matrix algebra by extending scalars, i.e. for a suitable field extension K of F, is isomorphic to the 2×2 matrix algebra over K.

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Linked quaternion algebras

Let F be a field of characteristic not equal to 2. Let A = (a1,a2) and B = (b1,b2) be quaternion algebras over F. The algebras A and B are linked quaternion algebras over F if there is x in F such that A is equivalent to (x,y) and B is equivalent to (x,z). [1] :69

In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive identity the ring is said to have characteristic zero.

The Albert form for A, B is

It can be regarded as the difference in the Witt ring of the ternary forms attached to the imaginary subspaces of A and B. [2] The quaternion algebras are linked if and only if the Albert form is isotropic. [1] :70

In mathematics, a quadratic form over a field F is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is anisotropic. More precisely, if q is a quadratic form on a vector space V over F, then a non-zero vector v in V is said to be isotropic if q(v) = 0. A quadratic form is isotropic if and only if there exists a non-zero isotropic vector for that quadratic form.

Linked fields

The field F is linked if any two quaternion algebras over F are linked. [1] :370 Every global and local field is linked since all quadratic forms of degree 6 over such fields are isotropic.

In mathematics, a global field is a field that is either:

In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology. Given such a field, an absolute value can be defined on it. There are two basic types of local fields: those in which the absolute value is Archimedean and those in which it is not. In the first case, one calls the local field an Archimedean local field, in the second case, one calls it a non-Archimedean local field. Local fields arise naturally in number theory as completions of global fields.

The following properties of F are equivalent: [1] :342

In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras over K, with addition given by the tensor product of algebras. It was defined by the algebraist Richard Brauer.

In mathematics, a biquaternion algebra is a compound of quaternion algebras over a field.

In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.

A nonreal linked field has u-invariant equal to 1,2,4 or 8. [1] :406

Related Research Articles

In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a nonzero ring in which every nonzero element a has a multiplicative inverse, i.e., an element x with a·x = x·a = 1. Stated differently, a ring is a division ring if and only if the group of units equals the set of all nonzero elements. A division ring is a type of noncommutative ring under the looser definition where noncommutative ring refers to rings which are not necessarily commutative.


In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing. They are named after the English mathematician William Kingdon Clifford.

In mathematics, the Cayley–Dickson construction, named after Arthur Cayley and Leonard Eugene Dickson, produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. The algebras produced by this process are known as Cayley–Dickson algebras, for example complex numbers, quaternions, and octonions. These examples are useful composition algebras frequently applied in mathematical physics.

In ring theory and related areas of mathematics a central simple algebra (CSA) over a field K is a finite-dimensional associative algebra A, which is simple, and for which the center is exactly K. As an example, note that any simple algebra is a central simple algebra over its center.

Square (algebra) result of multiplying a number, or other expression, by itself

In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the square of 3 may be written as 32, which is the number 9. In some cases when superscripts are not available, as for instance in programming languages or plain text files, the notations x^2 or x**2 may be used in place of x2.

In mathematics, the Hasse invariant of a quadratic form Q over a field K takes values in the Brauer group Br(K). The name "Hasse–Witt" comes from Helmut Hasse and Ernst Witt.

In mathematics, a composition algebraA over a field K is a not necessarily associative algebra over K together with a nondegenerate quadratic form N that satisfies

In mathematics, Witt's theorem, named after Ernst Witt, is a basic result in the algebraic theory of quadratic forms: any isometry between two subspaces of a nonsingular quadratic space over a field k may be extended to an isometry of the whole space. An analogous statement holds also for skew-symmetric, Hermitian and skew-Hermitian bilinear forms over arbitrary fields. The theorem applies to classification of quadratic forms over k and in particular allows one to define the Witt group W(k) which describes the "stable" theory of quadratic forms over the field k.

In mathematics, a Witt group of a field, named after Ernst Witt, is an abelian group whose elements are represented by symmetric bilinear forms over the field.

A non-associative algebra is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure A is a non-associative algebra over a field K if it is a vector space over K and is equipped with a K-bilinear binary multiplication operation A × A → A which may or may not be associative. Examples include Lie algebras, Jordan algebras, the octonions, and three-dimensional Euclidean space equipped with the cross product operation. Since it is not assumed that the multiplication is associative, using parentheses to indicate the order of multiplications is necessary. For example, the expressions (ab)(cd), d and a(b ) may all yield different answers.

In mathematics, a Pfister form is a particular kind of quadratic form, introduced by Albrecht Pfister in 1965. In what follows, quadratic forms are considered over a field F of characteristic not 2. For a natural number n, an n-fold Pfister form over F is a quadratic form of dimension 2n that can be written as a tensor product of quadratic forms

In mathematics, an octonion algebra or Cayley algebra over a field F is an algebraic structure which is an 8-dimensional composition algebra over F. In other words, it is a unital non-associative algebra A over F with a non-degenerate quadratic form N such that

In mathematics, the universal invariant or u-invariant of a field describes the structure of quadratic forms over the field.

In mathematics, a quadratically closed field is a field in which every element of the field has a square root in the field.

In mathematics, a quaternionic structure or Q-structure is an axiomatic system that abstracts the concept of a quaternion algebra over a field.

References

  1. 1 2 3 4 5 Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. 67. American Mathematical Society. ISBN   0-8218-1095-2. MR   2104929. Zbl   1068.11023.
  2. ↑ Knus, Max-Albert (1991). Quadratic and Hermitian forms over rings. Grundlehren der Mathematischen Wissenschaften. 294. Berlin etc.: Springer-Verlag. p. 192. ISBN   3-540-52117-8. Zbl   0756.11008.
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