唯一可除群中根可解的方程
Equations solvable by radicals in a uniquely divisible group
摘要：
研究了群G中的方程，对每个正整数m都有唯一的m次根.两个字母的词等式是形式w（X，A）= B的表达式，其中w是字母表{X，A}中的有限词。我们认为G中的A、B是固定系数，G中的X是未知数。某些字方程，如XAXAX=B，有根据根式的解决方案，而其他如XXAX = B没有。我们获得了第一个已知的无限家庭的字方程不能解决的自由基，并推测一个完整的分类。对于一个词w，我们将Z[x，y]中的一个多项式P_w与两个交换变量相关联，只要w是一个较小的词的组合，它就因式分解。证明了若P_w（x^2，y^2）在Z[x，y]中有绝对不可约因子，则方程w（X，A）=B不可用根解.
英文摘要：
We study equations in groups G with unique m-th roots for each positive integer m. A word equation in two letters is an expression of the form w(X,A) = B, where w is a finite word in the alphabet {X,A}. We think of A,B in G as fixed coefficients, and X in G as the unknown. Certain word equations, such as XAXAX=B, have solutions in terms of radicals, while others such as XXAX = B do not. We obtain the first known infinite families of word equations not solvable by radicals, and conjecture a complete classification. To a word w we associate a polynomial P_w in Z[x,y] in two commuting variables, which factors whenever w is a composition of smaller words. We prove that if P_w(x^2,y^2) has an absolutely irreducible factor in Z[x,y], then the equation w(X,A)=B is not solvable in terms of radicals.
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