1. ## fields and elements

Let K be a finite field with k elements. Let K*={x1,x2,...,xk-1}, the non-zero elements of K.
Show that product (x belonging to K*) x=x1x2...xk-1=-1.

Hint: Try to arrange, in the product, the elements in pairs a,b with ab=1, i.e. b=a^(-1) to the extent possible. Of course, a can not equal a^(-1) Which elements aren't paired?

I'm highly confused by this. I get it in the concrete example, i.e. in Z11, we get an arrangement of this form (2*6)(3*4)(5*9)(7*8) (1)(10), but I'm having trouble with it in the abstract.

Thanks.

Let K be a finite field with k elements. Let K*={x1,x2,...,xk-1}, the non-zero elements of K.
Show that product (x belonging to K*) x=x1x2...xk-1=-1.

Hint: Try to arrange, in the product, the elements in pairs a,b with ab=1, i.e. b=a^(-1) to the extent possible. Of course, a can not equal a^(-1) Which elements aren't paired?

I'm highly confused by this. I get it in the concrete example, i.e. in Z11, we get an arrangement of this form (2*6)(3*4)(5*9)(7*8) (1)(10), but I'm having trouble with it in the abstract.

Thanks.
DISCLAIMER: You may want to wait for verification by another member. I am NOT too experienced in field theory.

For every $\displaystyle k\in K-\{1,-1\}$ there exists some $\displaystyle k'$ such that $\displaystyle kk'=1$. Thus, $\displaystyle \prod_{k\in K}k=\prod_{k\in K-\{1,-1\}}k\cdot 1\cdot -1=(k_1k'_1)(k_2k'_2)\cdots =-1\cdot1\cdots=-1$