What are the units $x$ in $ℤ/Nℤ$ of the form $x = 1 + \overline{kd}$ for a divisor $d$ of $N$ and $k ∈ ℤ$, i.e. $$U_N[d] := \{x ∈ (ℤ/Nℤ)^×;\; ∃ k ∈ ℤ : x = 1 + \overline{kd}\} = \ker \big((ℤ/Nℤ)^× → (ℤ/dℤ)^×\big)?$$

  • How many such units are there?
  • How (else) do they look like?

The following is a wrong formulation and needs to be fixed:

Now, let $π_e$ denote the projection $ℤ/Nℤ → ℤ/eℤ$ for divisors $e$ of $N$. Then $U_N[d] = \ker π_d$. What I really want to know is:

  • Is $π_{N/d}(\ker{π_d})$ bijectively related to the units in $ℤ/(d,N/d)ℤ$?
  • $\begingroup$ To form the quotient $\,\Bbb Z/\Bbb N\,$ , what algebraic structure are you working with? Or you did mean $\,\Bbb Z/N\Bbb Z\,$ ? $\endgroup$ – DonAntonio May 31 '13 at 16:33
  • $\begingroup$ @DonAntonio Yes, I meant $ℤ/Nℤ$, the ordinary quotients of $ℤ$ by ideals $Nℤ ⊂ ℤ$. $\endgroup$ – k.stm May 31 '13 at 16:35

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