8
$\begingroup$

Let $f_{1},f_{2},\ldots,f_{n}$ be different automorphisms of field $\mathbb{K}$ . What I want to ask is:

Does there exist an element $x \in \mathbb{K}$ such that $f_{1}(x),f_{2}(x),\ldots,f_{n}(x)$ are pairwise distinct?

  • 1
    Hi, maths! Your last name seems familiar: haven't we met met before in some friendly circle?2012-08-16
  • 0
    I'd imagine the problem to be clearer if more restrictions were made on the field K. For example, whether K is Galois/algebraic/finite over its base field (Q or F_p)2012-08-16
  • 1
    Dear @Michael, the problem is perfectly clear and very interesting as stated. Your suggestion would only make it easier and/or less interesting.2012-08-16
  • 0
    Equivalently: Given $n$ automorphisms $f_1, \ldots, f_n$ different from the identity, prove that there is an element $x$ in $K$ such that $f_i(x) \neq x$ for all $f_i$.2012-08-16

2 Answers 2