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Let F be a finite field with n elements. Prove that $x^{n-1}=1$ for all nonzero x in F.

I'm not understanding where this proof is going. So we have a ring $Z_{7}$, and we know there are 7 elements in $Z_7$ which are $\{0,1,2,3,4,5,6\}$ (a complete system of residues modulo 7), obviously every nonzero element to the power of 6, (from n-1 where n=7) is congruent to 1 modulo 7. However, I'm not understanding what they want me to show. I could show this by induction but I get the feeling that there is an easier way to show this?

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7  
$F\setminus \{0\} $ is a multiplicative group. Use Lagrange's theorem. –  Ragib Zaman May 6 '12 at 7:28
3  
And you can't assume that $F$ is of the form $\Bbb Z_n$. –  Brian M. Scott May 6 '12 at 7:29
    
Just out of curiosity, how would you show it by induction? –  Tara B May 6 '12 at 10:06

3 Answers 3

up vote 3 down vote accepted

As noted by Ragib if $F$ is field then $F - \{0\}$ is a multiplicative group of order $n-1$. Therefore the order of any element $x$ in here must be a divisor of $n-1$, viz. if $m$ is the order of $x$, then $m | n-1$. Hence $qm = (n-1)$ for some $q \in \Bbb{N}$. It follows that

$$x^{n-1} = x^{mq} = (x^m)^q = 1^q = 1.$$

Q.E.D.

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A simple consequence of Lagrange's theorem is that for any finite group $G$:

$g^{|G|} = e$

for all $g\in G$.

In your case you have a multiplicative group of order $n-1$. Using the result we must have that $x^{n-1} = 1$ for all non-zero $x\in F$.

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The number of answers using Langrange's theorem is too damn high ! ;-)

So I will proceed differently. Let $x\in F$, $x\not= 0$. Then as $F$ is a field, the map $\alpha : y\mapsto x y$ is a bijection from $F^{\times}$ to $F^{\times}$. Therefore we have that $\Pi_{y\in F^{\times}} y = \Pi_{y\in F^{\times}} \alpha(y) = \Pi_{y\in F^{\times}} (xy) = x^{\textrm{Card}(F^{\times})} \Pi_{y\in F^{\times}} y$. Now, the element $\Pi_{y\in F^{\times}} y$ is non-zero because $F$ is a field, so that we get by simplification by $\Pi_{y\in F^{\times}} y$ that $x^{\textrm{Card}(F^{\times})} = 1$. Now, if $n$ is the cardinal of $F$, as $n - 1 = \textrm{Card}(F^{\times})$ because $F$ is a field, we get that $x^{n-1} = 1$, as required.

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This works in every finite abelian group. –  Martin Brandenburg Oct 31 at 22:01
    
Yes, I know, but no need to mention it, shush ! ;-) –  Robert Green Oct 31 at 22:02

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