# Does the maximum number of roots in a field directly imply the maximum number of solutions in a group

From Proposition 2.5 from https://wstein.org/edu/2007/spring/ent/ent-html/node28.html#prop:dsols, the maximum number of roots $$\alpha\in k$$ of $$x^n-1$$ in a field $$k$$ is $$n$$. That is, there are at most $$n$$ many $$\alpha$$ such that $$\alpha^n-1=0$$ in $$k$$.

I was wondering if it is true, and if so how to prove, that this maximum implies there are at most $$n$$ solutions to $$x^n=1$$ in the corresponding multiplicative group $$(k\backslash \{0\},\cdot)$$.

Logically, I would assume it does, as $$0$$ cannot be a root of $$x^n-1$$ in $$k$$, but I am very new to group theory, and have often found that my logic is wrong.

Thanks :)

• Precisely what do you doubt here? Likely you will get much more helpful answers if you make that clear. – Bill Dubuque Jul 13 at 18:53
• Assuming known that a quadratic has at most two rational roots can you deduce that it has at most two integer roots, without any such doubts? – Bill Dubuque Jul 13 at 19:02