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