Number Theory

Quadratic forms and their applications: proceedings of the by Conference on Quadratic Forms and Their Applications (1999 :

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By Conference on Quadratic Forms and Their Applications (1999 : University College Dublin), Eva Bayer-Fluckiger, David Lewis

This quantity outlines the complaints of the convention on "Quadratic varieties and Their purposes" held at collage collage Dublin. It contains survey articles and examine papers starting from functions in topology and geometry to the algebraic thought of quadratic kinds and its heritage. a number of facets of using quadratic types in algebra, research, topology, geometry, and quantity concept are addressed. detailed gains comprise the 1st released evidence of the Conway-Schneeberger Fifteen Theorem on integer-valued quadratic kinds and the 1st English-language biography of Ernst Witt, founding father of the speculation of quadratic kinds.

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Extra info for Quadratic forms and their applications: proceedings of the Conference on Quadratic Forms and Their Applications, July 5-9, 1999, University College Dublin

Example text

Therefore ϕU (V ) is a power of q. For any k = 0, . . , n there is a linear map σ ∈ GL(n, q) which has 1 as an eigenvalue of geometric multiplicity q k . If q is odd, choose the diagonal matrix Ek ⊕ (−En−k ). Let q be even. Then x → x + t(1, . . , 1) is a fixed point free involution. For k = 2, . . , n set   1 1 0    0 ...    ∈ Mk (Fq ) Jk =  .  . .. 1   .. 0 ... 0 1 and Bk = Jk ⊕ En−k . Then Bk is an element of 2-power order having q n−k+1 fixed points. Any involution has a Jordan matrix of the form r · J2 ⊕ En−2r .

7. Let the finite group G act on set S. Let n = S and set ϕ(S) = {ϕU (S) : U < G, ϕU (S) ≡ 0 mod 2}. Now define pG,S = (X − k). k∈ϕ(S) Set pG,H = pG,G/H . The result of Lewis and McGarraghy is as follows. 8. Let A = K[X]/(f (X)) be an ´etale K-algebra. Consider the action of the Galois group G of f (X) on the set S of roots of f (X). Then pG,S (X) annihilates the trace form < A >. As we will see, this theorem improves the Beaulieu-Palfrey result in two directions. It also holds for ´etale algebras and, as we see later, there are examples where the degree of pG,S (X) is strictly smaller then the degree of BG,H (X).

K∈{signσ (S):σ∈G,σ 2 =1} For S = G/H set qG,H (X) = qG,S . 11. Let H, G be as above. Then IM (G,H) ⊂ (qG,H (X)). There exists some integer e ≥ 0, which only depends on G and H, such that (2e qG,H (X)) ⊂ IM (G,H) . We call S σ a signature, since this value corresponds to signatures of trace forms via the homomorphism hN/K . 3. The main result on annihilating polynomials In this section we analyze the known vanishing results. 1. Let G be a finite group with Sylow 2-group G2 and let H < G be a subgroup of index n with ∩σ∈G σHσ −1 = 1.

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