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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 \(p\)

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  1. freckles
    • one year ago
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    I might be an idiot for saying this but I wonder if this has anything to do with fermat's little theorem

  2. ganeshie8
    • one year ago
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    Yes that \(x^{p-1}-1\) piece comes from little fermat :)

  3. freckles
    • one year ago
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    \[a_0=(-1)(-2)(-3) \cdots (-(p-1))+1 \\ \\ \text{ so } a_0=-(p-1)!+1 \\ \text{ so is } a_0 \text{ divisible by } p? \\ \\\] ...

  4. freckles
    • one year ago
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    just trying to look at a_0 right now because I thought that would be easiest

  5. ganeshie8
    • one year ago
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    it should be \(+(p-1)!+1\) right ? recall wilson

  6. freckles
    • one year ago
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    you are right p is odd so p-1 is even

  7. freckles
    • one year ago
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    p is odd for the most part*

  8. ganeshie8
    • one year ago
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    hey its not super human at all, its elementary number theory.. you will like it if you try few problems @Pikachubacca

  9. ganeshie8
    • one year ago
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    yes that gives us another way to prove wilson's thm http://mathworld.wolfram.com/WilsonsTheorem.html

  10. freckles
    • one year ago
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    I think I'm lost on the other coefficients.

  11. ganeshie8
    • one year ago
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    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
    • one year ago
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    it should p-1 is even so 2|(p-1) so p|a_(p-2)

  13. ganeshie8
    • one year ago
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    right, other coefficients look a bit complicated

  14. freckles
    • one year ago
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    so i guess this isn't the best route :p

  15. freckles
    • one year ago
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    I'm going to drink some coffee and eat some cookies I will be back later

  16. ganeshie8
    • one year ago
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    okay enjoy your coffee and snacks :)

  17. ikram002p
    • one year ago
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    idk should i ruin the fun or just give a hint :O

  18. ganeshie8
    • one year ago
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    first give hint maybe..

  19. ikram002p
    • one year ago
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    ok in Z_p of order x \(x^{p-1}-1= (x-1)(x-2)(x-3)\cdots (x-(p-1))\)

  20. ganeshie8
    • one year ago
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    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
    • one year ago
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    well its proof goes nice :O

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