## ganeshie8 one year ago consider the polynomial $f(x) = (x-1)(x-2)(x-3)\cdots (x-(p-1))-(x^{p-1}-1)\\=a_{p-2}x^{p-2}+a_{p-3}x^{p-3}+\cdots+a_1x+a_0$ show that each of the coefficient, $$a_i$$ is divisible by the prime $$p$$

1. freckles

I might be an idiot for saying this but I wonder if this has anything to do with fermat's little theorem

2. ganeshie8

Yes that $$x^{p-1}-1$$ piece comes from little fermat :)

3. freckles

$a_0=(-1)(-2)(-3) \cdots (-(p-1))+1 \\ \\ \text{ so } a_0=-(p-1)!+1 \\ \text{ so is } a_0 \text{ divisible by } p? \\ \\$ ...

4. freckles

just trying to look at a_0 right now because I thought that would be easiest

5. ganeshie8

it should be $$+(p-1)!+1$$ right ? recall wilson

6. freckles

you are right p is odd so p-1 is even

7. freckles

p is odd for the most part*

8. ganeshie8

hey its not super human at all, its elementary number theory.. you will like it if you try few problems @Pikachubacca

9. ganeshie8

yes that gives us another way to prove wilson's thm http://mathworld.wolfram.com/WilsonsTheorem.html

10. freckles

I think I'm lost on the other coefficients.

11. ganeshie8

not too sure if it is useful but it seems the coefficient of $$x^{p-2}$$ is given by \large{\begin{align}a_{p-2} &= (-1)+(-2)+(-3)+\cdots+(-(p-1)) \\~\\ &= -\dfrac{p(p-1)}{2}\\~\\ &=-\binom{p}{2} \end{align}}

12. freckles

it should p-1 is even so 2|(p-1) so p|a_(p-2)

13. ganeshie8

right, other coefficients look a bit complicated

14. freckles

so i guess this isn't the best route :p

15. freckles

I'm going to drink some coffee and eat some cookies I will be back later

16. ganeshie8

okay enjoy your coffee and snacks :)

17. ikram002p

idk should i ruin the fun or just give a hint :O

18. ganeshie8

first give hint maybe..

19. ikram002p

ok in Z_p of order x $$x^{p-1}-1= (x-1)(x-2)(x-3)\cdots (x-(p-1))$$

20. ganeshie8

right, both left hand side and right hand side are congruent to $$0\mod p$$ for all $$x\in\{1,2,3,\ldots,(p-1)\}$$ i don't really know how to use that hint

21. ikram002p

well its proof goes nice :O