Compositions of Quadratic Forms (De Gruyter Expositions in by Daniel B. Shapiro

By Daniel B. Shapiro

The principal subject of this booklet is the concept of Hurwitz and Radon bearing on composition formulation for sums of squares, first proved within the 1920's. ideas from algebra and topology are used to generalize that theorem in different instructions. The textual content contains labored examples and lots of workouts, which boost nonetheless extra diversifications of the vital subject. the most viewers is those who have had a few graduate classes in summary algebra, yet many sections of the booklet are available to a person with a few education in linear algebra. a number of significant subject matters during this booklet can be of curiosity to scholars of topology and geometry. the writer has tried to make the presen-tation as transparent and as trouble-free as attainable.

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Example text

Define f to be proper if det f = µ(f )m . The proper similarities form a subgroup Sim+ (V ) of index 2 in Sim• (V ). This is the analog of the special orthogonal group O+ (n) = SO(n). (3) Suppose f˜ = −f . If g = a1V + bf for a, b ∈ F then g is proper. (4) Wonenburger’s Theorem. Suppose f, g ∈ Sim• (V ) and f˜ = −f . If g commutes with f , then g is proper. If g anticommutes with f and 4 | n then g is proper. (5) Let L0 , R0 ⊆ Sim( 1, 1, 1, 1 ) be the subspaces described in Exercise 4(2). Let G be the group generated by L•0 and R0• .

If a, b are anisotropic, orthogonal elements of A◦ (using the norm form) then a 2 , b2 ∈ F • and ab = −ba. These elements might fail to generate a quaternion subalgebra. (4) There exists a basis e0 , e1 , . . , e2n −1 of A such that: e0 = 1 e1 , . . , e2n −1 ∈ A◦ pairwise anti-commute and ei2 ∈ F • . For each i, j : ei ej = λek for some index k = k(i, j ) and some λ ∈ F • . 1. Spaces of Similarities 35 The basis elements are “alternative”: ei ei · x = ei · ei x and xei · ei = x · ei ei . (5) The norm form x → [x] on A is the Pfister form −α1 , .

Then f is irreducible and if p ∈ F [x, Y ] vanishes on Z(f ) then f |p. 30 1. Spaces of Similarities Proof outline. Express p = c0 (Y )x d + · · · + cd (Y ). Since xg ≡ −h (mod f ), we have g d p ≡ Q (mod f ) where Q = c0 (Y )(−h(Y ))d + · · · + cd (Y )g(Y )d ∈ F [Y ]. Since p vanishes on Z(f ) it follows that Q(B) = 0 for every B ∈ F n with g(B) = 0. Therefore g · Q vanishes identically on F n . Hence Q = 0 as a polynomial and f |g d p and we conclude that f |p. (2) Suppose deg p = d and deg f = m above.

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