Hasse invariant of a quadratic form
Encyclopedia
In mathematics
, the Hasse invariant (or Hasse–Witt invariant) of a quadratic form
Q over a field
K takes values in the Brauer group
Br(K).The quadratic form Q may be taken as a diagonal form
Its invariant is then defined as the sum of the classes in the Brauer group of all the quaternion algebra
s
for i < j.
It may also be viewed as the second Stiefel-Whitney class of Q.
The name "Hasse–Witt" comes from Helmut Hasse
and Ernst Witt
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Hasse invariant (or Hasse–Witt invariant) of a quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
Q over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
K takes values in the Brauer group
Brauer group
In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras of finite rank over K and addition is induced by the tensor product of algebras. It arose out of attempts to classify division algebras over a field and is...
Br(K).The quadratic form Q may be taken as a diagonal form
- Σ aixi2.
Its invariant is then defined as the sum of the classes in the Brauer group of all the quaternion algebra
Quaternion algebra
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...
s
for i < j.
It may also be viewed as the second Stiefel-Whitney class of Q.
The name "Hasse–Witt" comes from Helmut Hasse
Helmut Hasse
Helmut Hasse was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of p-adic numbers to local classfield theory and diophantine geometry , and to local zeta functions.-Life:He was born in Kassel, and died in...
and Ernst Witt
Ernst Witt
Ernst Witt was a German mathematician born on the island of Als . Shortly after his birth, he and his parents moved to China, and he did not return to Europe until he was nine....
.