MWITT Witt rings of fields

Přírodovědecká fakulta
podzim 2010
Rozsah
0/0. 1 kr. (plus 1 za zk). Doporučované ukončení: z. Jiná možná ukončení: zk.
Vyučující
Prof. Kazimierz Szymiczek (přednášející), prof. RNDr. Radan Kučera, DSc. (zástupce)
Garance
prof. RNDr. Radan Kučera, DSc.
Ústav matematiky a statistiky – Ústavy – Přírodovědecká fakulta
Kontaktní osoba: prof. RNDr. Radan Kučera, DSc.
Omezení zápisu do předmětu
Předmět je otevřen studentům libovolného oboru.
Cíle předmětu
The course is intended to be an accessible introduction to the fundamental concept of the Witt ring of bilinear (or quadratic) forms over a field for a student who already knows the rudiments of groups, rings, fields and vector spaces.
Osnova
  • Part 1 of the course (4 hours) gives an introduction to the geometry of bilinear spaces and can be viewed as a continuation of the first course in linear algebra.
  • Part 2 of the course (6 hours) is devoted to the fundamental concept of the Witt ring of bilinear forms over an arbitrary field of characteristic not two. It offers a complete treatment of the most significant structural results for Witt rings.
  • Part 3 of the course (2 hours) explains the techniques of finite dimensional algebras and their applications to quadratic forms. This includes the construction of the Brauer group of a field and a discussion of two fundamental invariants: the Hasse invariant (of equivalence of quadratic forms) and the Witt invariant (of similarity of quadratic forms).
  • Part 1: Bilinear spaces
  • Bases and matrices of bilinear spaces.
  • Isometries of bilinear spaces.
  • Nonsingular bilinear spaces.
  • Diagonalization of bilinear spaces.
  • Witt's cancellation theorem.
  • Witt's chain isometry theorem.
  • Symmetric spaces over some fields.
  • Real and nonreal fields with small square class groups.
  • Classification of symmetric spaces.
  • Part 2: Witt rings
  • Hyperbolic spaces and Witt decomposition of symmetric spaces. The Witt group of a field.
  • Tensor products.
  • The Witt ring of a field.
  • Pfister forms.
  • Multiplicative properties.
  • The level of a nonreal field.
  • Witt ring of a nonreal field.
  • Formally real fields and ordered fields.
  • Prime ideals of the Witt ring.
  • Units and zero divisors in Witt rings.
  • Pythagorean fields.
  • Witt equivalence of fields.
  • Equivalence of fields with respect to quadratic forms.
  • Part 3: Invariants
  • Algebras.
  • Central simple algebras.
  • Hamilton quaternions.
  • Quaternion algebras.
  • Tensor product of algebras.
  • Internal direct product of subalgebras.
  • The Hasse algebra.
  • The reciprocal algebra.
  • Brauer group of a field.
  • Wedderburn's uniqueness theorem.
  • Hasse invariant.
  • Witt invariant.
  • Arason-Pfister property.
  • Harrison's criterion.
Literatura
  • SZYMICZEK, Kazimierz. Bilinear algebra. An introduction to the algebraic theory of quadratic forms. Amsterdam: Gordon and Breach Science Publishers, 1997, 486 s. ISBN 90-5699-076-4. info
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Předmět je vyučován jednorázově.
Výuka probíhá blokově.
Předmět je zařazen také v obdobích podzim 2010 - akreditace, podzim 2011 - akreditace.
  • Statistika zápisu (nejnovější)
  • Permalink: https://is.muni.cz/predmet/sci/podzim2010/MWITT