We prove a result on the asymptotic proportion of randomly chosen pairs \((\sigma ,\tau )\) of permutations in the symmetric group \(S_n\) which “invariably” generate a nonsolvable subgroup, i.e., whose cycle structures cannot possibly both occur in the same solvable subgroup of \(S_n\). As an application, we obtain that for a large degree “random” integer polynomial f, reduction modulo two different primes can be expected to suffice to prove the nonsolvability of \(\text {Gal}(f/{\mathbb {Q}})\).

中文翻译：

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关于不可解置换组的不变生成的注记

我们证明了对称组\（S_n \）中随机选择的置换对\（（\ sigma，\ tau）\）的渐近比例的结果，对称组\（S_n \） “总是”生成一个不可解的子组，即，其循环结构不可能两者都出现在\（S_n \）的同一可解决子组中。作为一个应用，我们获得了对于一个很大程度的“随机”整数多项式f，可以期望对两个不同素数进行模减，足以证明\（\ text {Gal}（f / {\ mathbb {Q}}的不可解性）\）。