※ 引述《wuxr (wuxr)》之銘言：
: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=61&t=468974
: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=61&t=468846
: 求教了 謝謝^^
我大概給個方向, 其他的可以自己試試看.
Q1. (Case for charF=p)
(=>) If x^p-c has a root in F, say r. Then x^p-c=x^p-r^p=(x-r)^p,
which is reducible over F.
(<=) Let r be a root of x^p-c (in some field extension E of F.)
Let q(x)=irr(r,F), then q|(x-r)^p in E[x].
Write q(x)=(x-r)^k for some integer k.
Your goal is to show that k=p.
Q2.
(a)
(=>) It is similar to the argument of (=>) in Q1.
(<=) Note that if r is a root of x^p-x-c, then the other ones are exactly
r+1, r+1+1, ..., r+(p-1)1.
Now you should claim that irr(r,F)=x^p-x-c by comparing the
coefficient of x^(p-1).
Good luck!
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