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 |
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Wiley
2000-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171200002830 |
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| _version_ | 1832546511326543872 |
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| author | Javier Gomez-Calderon |
| author_facet | Javier Gomez-Calderon |
| author_sort | Javier Gomez-Calderon |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-1d83842833094812b5a2c35334ef545c |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2000-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-1d83842833094812b5a2c35334ef545c2025-02-03T06:48:35ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252000-01-01231177778110.1155/S0161171200002830On the decomposition of xd+aexe+⋯+a1x+a0Javier Gomez-Calderon0Department of Mathematics, New Kensington Campus, Pennsylvania State University, New Kensington 15068, PA, USALet 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.http://dx.doi.org/10.1155/S0161171200002830Polynomials and fields. |
| spellingShingle | Javier Gomez-Calderon On the decomposition of xd+aexe+⋯+a1x+a0 International Journal of Mathematics and Mathematical Sciences Polynomials and fields. |
| title | On the decomposition of xd+aexe+⋯+a1x+a0 |
| title_full | On the decomposition of xd+aexe+⋯+a1x+a0 |
| title_fullStr | On the decomposition of xd+aexe+⋯+a1x+a0 |
| title_full_unstemmed | On the decomposition of xd+aexe+⋯+a1x+a0 |
| title_short | On the decomposition of xd+aexe+⋯+a1x+a0 |
| title_sort | on the decomposition of xd aexe ⋯ a1x a0 |
| topic | Polynomials and fields. |
| url | http://dx.doi.org/10.1155/S0161171200002830 |
| work_keys_str_mv | AT javiergomezcalderon onthedecompositionofxdaexea1xa0 |