Potenzieren und Wurzelziehen in rationalen Quaternionenalgebren |
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Authors: | G Kuba |
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Institution: | 1. Institut für Mathematik, Universit?t für Bodenkultur, Gregor Mendel-Stra?e 33, 1180, Wien, Austria
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Abstract: | LetF be a field not of characteristic 2 andQ =F +F
i +F
j +F
k the quaternion algebra overF whereij = -ji =k andi
2 = α andj
2 = β with 0 ≠ α, β ∈F fixed. (IfF = ℝ and α = β = - 1 thenQ is the division algebra of the Hamilton quaternions.) IfF = ℚ and Q is a division algebra then by embedding certain quadratic number fields inQ we derive an efficient formula to compute the powers of any quaternion. This formula is even true in general and reads as
follows. If a, a1, a2, a3 ∈F andn ∈ ℕ then
where ω ig a square root of αa1
2 + βa
2
2 - αβa
3
2 in or overF and
andA
0 =na
n-1.
With the help of this formula and related ones we are able to solve the equationX
n
=q for arbitrary quaternionsq and positive integers n in case ofF = ℝ and hence in case ofF ⊂ ℝ as well. IfF = ℝ then the total number of all solutions equals 0, 1, 2, 4,n or ∞. (4 is possible even whenn < 4.) In case ofF = ℚ, which we are primarily interested in, there are always either at most six or infinitely many solutions. Further, for
everyq ≠ 0 there is at most one solution provided thatn is odd and not divisible by 3. The questions when there are infinitely many solutions and when there are none can always
be decided by checking simple conditions on the radicandq ifF = ℝ. ForF = ℚ the two questions are comprehensively investigatet in a natural connection with ternary and quaternary quadratic rational
forms. Finally, by applying some of our theorems on powers and roots of quate-rions we also obtain several nice results in
matrix theory. For example, for every k ∈ ℤ the mappingA ↦A
k
on the group of all nonsingular 2-by-2 matrices over ℚ is injective if and only ifk is odd and not divisible by 3.
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Keywords: | 2000 Mathematics Subject Classification" target="_blank">2000 Mathematics Subject Classification 11R52 15A24 |
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