On the decomposition of xd+aexe+⋯+a1x+a0
Let K denote a field. A polynomial f(x)∈K[x] is said to be decomposable over K if f(x)=g(h(x)) for some polynomials g(x) and h(x)∈K[x] with 1<deg(h)<deg(f). Otherwise f(x) is called indecomposable. If f(x)=g(xm) with m>1, then f(x) is said to be trivially decomposable. In this paper, we sh...
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| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
2000-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171200002830 |
| Summary: | Let K denote a field. A polynomial f(x)∈K[x] is said to be
decomposable over K if f(x)=g(h(x)) for some
polynomials g(x) and h(x)∈K[x] with
1<deg(h)<deg(f).
Otherwise f(x) is called indecomposable. If
f(x)=g(xm) with m>1, then f(x) is said to be
trivially decomposable. In this paper, we show that xd+ax+b is
indecomposable and that if e denotes the largest proper divisor
of d, then xd+ad−e−1xd−e−1+⋯+a1x+a0 is either
indecomposable or trivially decomposable. We also show that if
gd(x,a) denotes the Dickson polynomial of degree d and
parameter a and gd(x,a)=f(h(x)), then
f(x)=gt(x−c,a) and h(x)=ge(x,a)+c. |
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| ISSN: | 0161-1712 1687-0425 |