Nonreal Field Research Articles

Supreme torsion forms

We study formally real, non-pythagorean fields which have an anisotropic torsion form that contains every anisotropic torsion form as a subform. We obtain consequences for certain invariants and the Witt ring of such fields and construct examples. We obtain a theory analogous to the theory of supreme Pfister forms introduced by Karim Becher and see examples in which the Pythagoras number for formally real fields behaves like the level for nonreal fields.

Triple linkage of quadratic Pfister forms

Given a field F of characteristic 2, we prove that if every three quadratic n-fold Pfister forms have a common quadratic $$(n-1)$$ -fold Pfister factor then $$I_q^{n+1} F=0$$ . As a result, we obtain that if every three quaternion algebras over F share a common maximal subfield then u(F) is either 0, 2 or 4. We also prove that if F is a nonreal field with $${\text {char}}(F) \ne ~2$$ and $$u(F)=4$$ , then every three quaternion algebras share a common maximal subfield.

Lower Bounds and Infinity Criterion for Brauer p‑dimensions of Finitely-generated Field Extensions

Let E be a field, p a prime number and F/E a finitely-generated extension of transcendency degree t. This paper shows that if the absolute Galois group GE is of nonzero cohomological p-dimension cdp(E), then the field F has Brauer p-dimension Brdp(F) ≥ t except, possibly, in case p = 2, the Sylow pro-2-subgroups of GE are of order 2, and F is a nonreal field. It announces that Brdp(F) is infinite whenever t ≥ 1 and the absolute Brauer p-dimension abrdp(E) is infinite; moreover, for each pair (m,n) of integers with 1 ≤ m ≤ n, there exists a central division F-algebra of exponent p m and Schur index p n .

Remark on rigidity over several fields

It is shown that T -spectrum representable cohomology theories on smooth algebraic varieties satisfy normalization condition over nonreal fields. As a consequence, one can see that the rigidity property holds for all representable theories over considered fields.

On the residue fields of Henselian valued stable fields

Let ( K , v ) be a Henselian valued field satisfying the following conditions, for a given prime number p: (i) central division K-algebras of (finite) p-primary dimensions have Schur indices equal to their exponents; (ii) the value group v ( K ) properly includes its subgroup p v ( K ) . The paper shows that if K ˆ is the residue field of ( K , v ) and R ˆ is an intermediate field of the maximal p-extension K ˆ ( p ) / K ˆ , then the natural homomorphism Br ( K ˆ ) → Br ( R ˆ ) of Brauer groups maps surjectively the p-component Br ( K ˆ ) p on Br ( R ˆ ) p . It proves that Br ( K ˆ ) p is divisible, if p > 2 or K ˆ is a nonreal field, and that Br ( K ˆ ) 2 is of order 2 when K ˆ is formally real. We also obtain that R ˆ embeds as a K ˆ -subalgebra in a central division K ˆ -algebra Δ ˆ if and only if the degree [ R ˆ : K ˆ ] divides the index of Δ ˆ .

Sums of squares of linear forms

Let k be a real field. We show that every non-negative homogeneous quadratic polynomial f (x(1),..., x(n)) with coefficients in the polynomial ring k[t] is a sum of 2n center dot tau(k) squares of linear forms, where tau(k) is the supremum of the levels of the finite non-real field extensions of k. From this result we deduce bounds for the Pythagoras numbers of affine curves over fields, and of excellent two-dimensional local henselian rings.

Supreme Pfister Forms#

For a nonreal field F of characteristic different from 2, we compare several properties which F may have with respect to an anisotropic n-fold Pfister form π over F. In particular, we consider the situation where the subforms of π are all the anisotropic quadratic forms over F; then π is said to be supreme. We further apply our results to the case where π is the form 2 n × ⟨1⟩ (n ≥ 1). In this way we obtain new examples of nonreal fields with prescribed level and with additional properties.

Stark Effect Revisited

We extend the rigorous theory of complex scaling to atoms in constant electric field. This allows one to give a precise mathematical definition of resonance and leads to several results about the perturbation series: Borel summability at nonreal field and a relation between the asymptotics of the perturbation coefficients for large $n$ and the width of the resonance for small field.

Kaplanaky's radical and quadratic forms over non-real fields

Kaplanaky's radical and quadratic forms over non-real fields

Structure of Witt rings, quotients of abelian group rings and orderings of fields

In particular, Theorems 5 and 6 yield the results of [5, §3] for Witt rings of formally real fields and Theorem 7 those of [5, §5] for Witt rings of nonreal fields. By studying subrings of the rings described in Theorems 5-7 and using the results of [2] for symmetric bilinear forms over a Dedekind ring