1
$\begingroup$

let $\zeta_5$ be the $5-$root of unit and $\lambda = 1-\zeta_5 $ I need to simplify this congruence $n\equiv 1 \pmod{\lambda^5}$ to be modulo an integer in $\Bbb Z$, is that possible?

$\endgroup$
2
$\begingroup$

The ideal generated by $1-\zeta_p$ in $\Bbb Z[\zeta_p]$ ($p$ prime) satisfies $(1-\zeta)^{p-1}=(p)$. Therefore for $p=5$ $$(5)\supset (1-\zeta_5)^5=(1-\zeta)(5)\supset (25)$$ with both containments strict. Therefore, for $n\in\Bbb Z$, $n\equiv1\pmod{(1-\zeta_5)^5}$ iff $n\equiv1\pmod{25}$.

$\endgroup$
  • $\begingroup$ thanks a lot my friend, one more question, can we have equivalence between the two congruence ?? $\endgroup$ – Fouad El Sep 22 at 12:16

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.