一个类天鹅定理
A Swan-like Theorem
摘要：
Richard G.斯旺在1962年证明了三项式x^{8 k} + x^m + 1且8 k> m有一个偶数不可约因子，因此不能是不可约的。事实上，他发现奇偶性的一些不可约的因素，任何平方免费三项式在F_2[x]。我们证明了一个结果，是类似的精神。即，设n是奇数，f（x）= x^n + Sum_{i in S} x^i + 1在F_2[x]中，其中S子集{i：i奇数，i < n/3}并{i：i = n（mod 4），i < n}证明了若n = +-1（mod 8）则f（x）有奇数个不可约因子，如果n = +=3（mod 8），则f（x）有偶数个不可约因子。这有一个应用程序的问题找到多项式基地{1，a，a^2，. F_{2^n}的a^{n-1}}，使得对于所有1 <= i < n，Tr（a^i）= 0。
英文摘要：
Richard G. Swan proved in 1962 that trinomials x^{8k} + x^m + 1 with 8k > m have an even number of irreducible factors, and so cannot be irreducible. In fact, he found the parity of the number of irreducible factors for any square-free trinomial in F_2[x]. We prove a result that is similar in spirit. Namely, suppose n is odd and f(x) = x^n + Sum_{i in S} x^i + 1 in F_2[x], where S subset {i : i odd, i < n/3} Union {i : i = n (mod 4), i < n} We show that if n = +-1 (mod 8) then f(x) has an odd number of irreducible factors, and if n = +=3 (mod 8) then f(x) has an even number of irreducible factors. This has an application to the problem of finding polynomial bases {1,a,a^2,...a^{n-1}} of F_{2^n} such that Tr(a^i) = 0 for all 1 <= i < n.
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