Classes of forms Witt equivalent to a second trace form over fields of characteristic two

Let F be a field of characteristic two. We determine all non-hyperbolic quadratic forms over F that are Witt equivalent to a second trace form.


Introduction
Let E/F be a finite separable field extension. We define the trace form for this extension by q(x) = tr E/F (x 2 ). When the characteristic of F is not equal to 2, the trace form (E, q) is non-degenerate. However, if the characteristic is 2 then (E, q) is degenerate and splits as [1] ⊥ V , with V totally isotropic. It is therefore natural to introduce a modified "second trace form". To this end one considers for each a ∈ E its characteristic polynomial p(x, a) = x n − T 1 (a)x n−1 + T 2 (a)x n−2 + · · · + (−1) n T n (a) (1) (whence T 1 (a) = tr E/F (a) and T n (a) is the norm of a). It is clear that (E, T 2 ) is a quadratic form. When the degree n of the extension is odd this form is necessarily singular. To arrive at a non-degenerate form, two methods have been proposed in the literature. One method, due to Bergé and Martinet [BM], increases the dimension of the space by 1 using theétale F -algebra. The other method, due to Revoy [R], reduces the dimension of the space by 1. In this note we will adopt the second method and call such forms 2-trace forms. We consider the following problem: Which elements [q] = 0 of the Witt-group W q (F ) are represented by 2-trace forms?. Our theorem 3 fully answers this question. Moreover, we will partially answer the same question for [q] = 0 (see Prop. 1 and Prop. 2). For fields of characteristic not equal to 2 this problem seems quite more complicated: for partial results concerning generic fields one may consult [CP] and [EHP]; a complete solution for Hilbertian fields is given in [Sch], [KS] and [Wat].

The second trace form
As we remarked in the introduction, there are two ways to define a second trace form. In this section we will prove that in fact the corresponding forms are Witt equivalent.
Let E/F be a finite separable field extension. The second trace form T E/F of the extension E/F was defined by Revoy [R] as (E, T 2 ) if the degree [E : F ] is even, and as (E 0 , T 2 ) if the degree is odd, where T 1 , T 2 are given by (1) and E 0 = Ker T 1 .
It is important to remark that the bilinear form b q associated to T E/F satisfies the following relations: On the other hand, Bergé and Martinet defined in [BM] the second trace form as Revoy if the degree [E : F ] is even and as (E × F, T 2 ) if not. Proof. If [E : F ] is odd, then the form (E × F, T 2 ) of Bergé and Martinet [BM, p. 14] splits as follows (F (1, 0) + F (0, 1)) ⊥ E 0 × {0}. Since F (1, 0) + F (0, 1) is an hyperbolic plane, the claim follows immediately. ✷

2-algebraic forms
In this section we determine all non-hyperbolic quadratic forms over F that are Witt equivalent to some second trace form. Furthermore we give fields where hyperbolic forms are Witt equivalent to a second trace form.
Proof. The assertion is deduced from Theorem 1 above and Theorem 3.5 in [BM,. In order to illustrate the ideas, we will give the proof in case [E : F ] = 2n + 1.

CLASSES OF FORMS WITT EQUIVALENT TO A SECOND TRACE FORM OF FIELDS 3
Corollary 2. If a non hyperbolic quadratic form (V, q) over F is 2-algebraic then there exists a quadratic extension field E of F such that the extension (V ⊗ F E, q E ) is hyperbolic.
Proof. We only need to find an extension E of F such that T E/F is hyperbolic. We first remark that p(x) := x 4 + x 3 + 1 is irreducible over F 2 and also over F 2 (t). Let α be a root of p and E = F (α). We decompose the trace form T E/F with respect to the basis {α, 1 + α 3 } ∪ {α 2 , α + α 2 + α 3 }. Noting that this basis has the elements conjugate to α, it is easy to recognise that each vector basis is isotropic, and furthermore by (2) we see that it is a symplectic basis. Hence, the space is hyperbolic. ✷ Theorem 4. Let F be a field. If there exists a ∈ F * and n odd such that the polynomial x n − a is irreducible over F [x], then hyperbolic quadratics space over F are 2-algebraic.