Permutations in a finite field
Permutations in a finite field
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有限域中的排列
DOI：
10.1090/s0002-9939-1953-0055965-8
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发表时间：
1953
期刊：
影响因子：
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通讯作者：
L. Carlitz
中科院分区：
文献类型：
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作者：
L. Carlitz
(1)ax+f，xq-2(at：EGF(Q)，a#0)。当q=5时，Betti证明了这一点；当q=7时，相应的结果被Dickson[1，第119页]所证实。在这篇注记中，我们很简单地证明了这一结果对所有的q都成立。由于置换多项式的全集明显地提供了q个字母上的对称群的表示，它将足以证明每个换位(GA)可以借助于特殊的多项式(1)来生成；这里a表示一个固定的非零数CGF(Q)。我们考虑下面的多项式
(1) ax + f, xQ-2 (at: EGF(q), a # 0). For q = 5, this was proved to be true by Betti; for q =7 the corresponding result was verified by Dickson [1, p. 119]. In this note we show very simply that this result holds for all q. Since the totality of permutation polynomials evidently furnishes a representation of the symmetric group on q letters, it will suffice to show that every transposition (Ga) can be generated by means of the special polynomials (1); here a denotes a fixed nonzero number CGF(q). We consider the following polynomial