The eigenvalues of four-dimensional rotations

Edwards, Bruce H.

doi:10.1080/03081087808817208

Linear and Multilinear Algebra

1978

DOI: 10.1080/03081087808817208

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The eigenvalues of four-dimensional rotations

Bruce H. Edwards

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1980

1983

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“…From [2] we know the roots of p(x) are of the form a, l/a, b, l/b and that Gal(A"/F) is the Klein 4-group. Note that one of a -a~x + b -b~x or a -a~x -b + b~x is nonzero, since otherwise a -a~x -0.…”

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Rotations and linkage of 2-fold Pfister forms

Fitzgerald¹

1983

Proc. Amer. Math. Soc.

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Abstract. We show that a pair of 2-fold Pfister forms admit rotations with the same irreducible, separable characteristic polynomial if and only if they are linked.Baeza [1] has described the quadratic forms over a field F which admit a rotation with a given irreducible, separable characteristic polynomial. Such forms arise as the Scharlau transfer of some binary quadratic form. It is natural to ask. as Baeza does, for the relation between two quadratic forms arising this way. Here we give the answer in the case char F =£ 2 and the forms are 2-fold Pfister forms. Namely, a pair of 2-fold Pfister forms admit rotations with the same irreducibile, separable characteristic polynomial if and only if they are linked. A partial result for other four-dimensional forms is also given.We follow the notation of [6] for quadratic forms. The underlying field F has characteristic different from 2 and all forms are nonsingular. The construction of the Scharlau transfer and its elementary properties are in [6]. We will use these results without further mention.A 2-fold Pfister form is the norm form of a quaternionic algebra; thus it can be denoted as Let O(q) denote the orthogonal group of the quadratic form q and let Oir(q) denote those elements of O(q) with irreducible, separable characteristic polynomials. Note that if the dimension of q is a 2-power, separability is implied by irreducibility. If A E OlT(q), let F(A) denote the splitting field of the charateristic polynomial of A.In the case we are interested in, namely q a 2-fold Pfister form, Edwards [2] has shown the Galois group of F(A)/F is the Klein 4-group, for all A E Oit(q).We review Baeza's construction. Let q be an even-dimensional form and A E Oir(q). LetpA(x) be the characteristic polynomial of A. Choose a root a of pA(x) and set L -F(a + a~x) and

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“…In the case we are interested in, namely q a 2-fold Pfister form, Edwards [2] has shown the Galois group of F(A)/F is the Klein 4-group, for all A E Oit(q).…”

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Rotations and linkage of 2-fold Pfister forms

Fitzgerald¹

1983

Proc. Amer. Math. Soc.

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Rotations and discriminants of quadratic spaces

Edwards¹

1980

Linear and Multilinear Algebra

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The eigenvalues of four-dimensional rotations

Cited by 2 publications

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Rotations and linkage of 2-fold Pfister forms

Rotations and linkage of 2-fold Pfister forms

Rotations and discriminants of quadratic spaces

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