It turns out though, if you think about the group which permutes the roots of such a polynomial, there's something about the nature of this group that reveals that no quintic formula can exist.

事实证明， 如果你考虑一下置换这种多项式根的群，这个群的性质揭示了不次方程的求根公式。

And that's because of all of the different patterns we can permute.

这是因为我可以改变所有不同的模式。

This particular isomorphism between cube rotations and permutations of four objects is a bit subtle, but for the curious among you, you may enjoy taking a moment to think hard about how the rotations of a cube permute its four diagonals.

这种立方体旋转与四个对象排列间的同构关系有点微妙，但对于你中好奇的人来说， 你可能会喜欢花点时间思考一下立方体的旋转是如何对其四条对角线进行排列的。

That however, would grossly overcount the  total number since all the different ways you can permute these four items would be  counted separately.

然而，这样会严重高估总，因为这四种项目的不同排列方式会被单独计。