Elementary Theory of Numbers – Polynomials Inducing the Zero Function mod $ n $

  • Which polynomials induce the zero function mod $ n $?

In particular:

These are not empty questions because of the following general result:

Yes $ r $ is the maximum exponent in the factorization in prime numbers of $ n $then $ x mapsto x ^ {r + lambda (n)} – x ^ r $ is the zero function mod $ n $. (Wikipedia)

Right here, $ lambda $ is the function of Carmichael.

  • When is $ x ^ {r + lambda (n)} – x ^ r $ the monique polynomial of less degree that induces the zero function mod $ n $?

Fermat's theorem implies that $ x ^ n-x $ is the answer for $ n $ prime: all the polynomials inducing the zero function mod $ n $ are a multiple of $ x ^ n-x $.

Here are some other examples:
$$
begin {array} {rll}
n & text {minimum degree} & text {minimum degree}
\ 2 & x ^ 2 + x
\ 3 & x ^ 3-x
\ 4 & 2 (x ^ 2 + x) & x ^ 4-x ^ 2
\ 5 & x ^ 5-x
\ 6 & 3 (x ^ 2 + x) & x ^ 3-x
\ 7 & x ^ 7-x
\ 8 & 4 (x ^ 2 + x) & x ^ 5 + 2x ^ 3 + 3x ^ 2 + 2x = x (x + 1) (x ^ 2 + x + 2)
\ 9 & 3 (x ^ 3-x) & x ^ 8-x quad (???)
\ 10 & 5 (x ^ 2 + x) & x ^ 5-x
\ 11 & x ^ {11} -x
\ 12 & 6 (x ^ 2 + x) & x ^ 4 + 5x ^ 2 + 6x = x (x + 1) (x ^ 2-x + 6)
\ 13 & x ^ {13} -x
\ 14 & 7 (x ^ 2 + x) & ???
\ 15 & 5 (x ^ 3-x) & x ^ 5-x
\ 21 & 7 (x ^ 3-x) & ???
\ 24 & 12 (x ^ 2 + x) & x ^ 4 + 2x ^ 3 + 11x ^ 2 + 10x = x (x + 1) (x ^ 2 + x + 10)
end {array}
$$