Let f(x) = x^n + (a[n-1] t + b[n-1]) x^(n-1) + ... + (a[0] t + b[0]) be of constant degree n in x and degree <= 1 in t, where all a[i],b[i] are randomly and uniformly selected from a finite field GF(q) of q elements. Then the probability that the Galois group of f over the rational function field GF(q)(t) is the symmetric group S(n) on n elements is 1 - O(1/q). Furthermore, the probability that the Galois group of f(x) over GF(q)(t) is not S(n) is >= 1/q for n >= 3 and > 1/q - 1/(2q^2) for n = 2.
翻译：Lets f(x) = x ⁇ n + (a[n-1] t + b[n-1]) x ⁇ (n-1) +... + (a[0] t + b[0]) 以恒定度n x 和°% 1 t, 此处所有 a(i), b(一) 均随机和统一地从q元素的有限字段的 GF(q) 中选择。 然后, Glois 组合的 f 在合理函数字段 GF(q) (t) 上的对称组 S(n) 是 1 - O(1/q) +...+. (a[0] t + b[0]) 在 x 和°% 1 和° 1 (t) 不是 S(n) 是 n 3 和 > 1/q 1/(1/2(2q) 2) = 2的 Glois grois group of f(x) over GF(q)(q) is s (n) is 1/q) is n= 2。
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