Linkage of Fields in Characteristic 2

Formally real fields from a Galois theoretic perspective

Powers of the Fundamental Ideal in the Witt Ring

Sur les dimensions des formes quadratiques de hauteur 2

On splitting the Knebusch–Milnor exact sequence

The generic quadratic form of even dimension n with trivial discriminant over an arbitrary field of characteristic different from 2 containing a square root of -1 can be written in the Witt ring as a sum of 2-fold Pfister forms using n -2 terms and not less.The number of 2-fold Pfister forms needed to express a quadratic form of dimension 6 with trivial discriminant is determined in various cases.

Let WF denote the Witt ring of a field F of characteristic ≠2 and let I n F denote the n-th power of the ideal IF of even-dimensional forms in WF. The Arason-Pfister Hauptsatz states that if 0≠ϕ∈I n F is anisotropic then dim ϕ≥ 2 n . Pfister also showed that if ϕ∈I 3 F is anisotropic and dim ϕ>8 then dim ϕ≥12. We extend this result to I 4 F and show that if ϕ∈I 4 F is anisotropic and dim ϕ>16 then dim ϕ≥24 and we provide some results on anisotropic 24-dimensional forms in I 4 F.

We provide a new proof of the Arason–Pfister Hauptsatz (APH) in the setting of reduced special groups, as developed by Dickmann and Miraglia. Our approach avoids the use of Boolean invariants and instead relies on a construction inspired by Marshall’s quotient, suitably adapted to the context of special groups. We establish structural properties of this quotient and show that it generalizes the Pfister quotient by a Pfister subgroup. Using this framework, we define iterated quadratic extensions of special groups and develop a theory of Arason–Pfister sequences. These tools allow us to prove that any anisotropic form φ∈In(G) over a reduced special group G satisfies the inequality dim(φ)≥2n, where In(G) denotes the n-th power of the fundamental ideal of the Witt ring of G. Our methods are purely algebraic and internal to the theory of special groups, contributing with novel tools to the categorical study of abstract theories of quadratic forms.

Nonstandard quadratic forms over rational function fields

The u-invariant of a field F, u = u(F), is defined to be the maximum of the dimensions of anisotropic quadratic forms over F. If F is a non-formally real field with a finite number q of square classes then it is known that u ≦ q. The purpose of this note is to give some necessary and sufficient conditions for equality in terms of the structure of the Witt ring of F.

Isotropy of Six-Dimensional Quadratic Forms over Function Fields of Quadrics

It seems that any deeper investigation of the algebraic theory of quadratic forms requires a thorough knowledge of the theory of Pfister forms. They are the subject of this chapter. The investigation of Pfister forms is based essentially on the use of transcendental extensions of the ground field: One considers quadratic forms Σ a ij X i X j . with indeterminates X i . This approach leads naturally to the important concept of the function field of a quadratic form which is the function field of the quadratic Σ a ij X i X j =0.

A field F F of characteristic ≠ 2 \neq 2 is said to have finite Pythagoras number if there exists an integer m ≥ 1 m\geq 1 such that each nonzero sum of squares in F F can be written as a sum of ≤ m \leq m squares, in which case the Pythagoras number p ( F ) p(F) of F F is defined to be the least such integer. As a consequence of Pfister’s results on the level of fields, p ( F ) p(F) of a nonformally real field F F is always of the form 2 n 2^n or 2 n + 1 2^n+1 , and all integers of such type can be realized as Pythagoras numbers of nonformally real fields. Prestel showed that values of the form 2 n 2^n , 2 n + 1 2^n+1 , and ∞ \infty can always be realized as Pythagoras numbers of formally real fields. We will show that in fact to every integer n ≥ 1 n\geq 1 there exists a formally real field F F with p ( F ) = n p(F)=n . As a refinement, we will show that if n , m ≥ 2 n,m\geq 2 and k ≥ 1 k\geq 1 are integers such that 2 m ≥ 2 k ≥ n 2m\geq 2^{k}\geq n , then there exists a uniquely ordered field F F with p ( F ) = n p(F)=n and u ( F ) = u ~ ( F ) = 2 m u(F)=\tilde {u}(F)=2m (resp. u ( F ) = u ~ ( F ) = ∞ u(F)=\tilde {u}(F)=\infty ), where u u (resp. u ~ \tilde {u} ) denotes the supremum of the dimensions of anisotropic forms over F F which are torsion in the Witt ring of F F (resp. which are indefinite with respect to each ordering on F F ).

We present here two new criteria for existence of a tame Harrison map of two formally real algebraic function fields over a fixed real closed field of constants. The first criterion (c.f. Theorem 2.5) shows that a square class group isomorphism is a tame Harrison map if it induces an isomorphism of the coproduct rings of residue Witt rings. The other result (c.f. Proposition 3.5) associates a tame Harrison map to an integral quaternion-symbol equivalence.

Isotropy over function fields of Pfister forms