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Solve-in-Z-X-1-XP-1-mod-X-4-1-2-X-3-P-P-1-X-2-mod-X-4-1-3-P-2-X-3-P-X-2-0-mod-X-2-2-




Question Number 146108 by savitar last updated on 11/Jul/21
 Solve  in Z[X]  1) XP ′ ≡ −1 mod(X^4 +1)  2) X^3 P −P ′ ≡ 1−X^2  mod(X^4 +1)  3) P^2 −X^3 P−X^2   ≡ 0 mod(X^2 +2)
$$\:{Solve}\:\:{in}\:\mathbb{Z}\left[{X}\right] \\ $$$$\left.\mathrm{1}\right)\:{XP}\:'\:\equiv\:−\mathrm{1}\:{mod}\left({X}^{\mathrm{4}} +\mathrm{1}\right) \\ $$$$\left.\mathrm{2}\right)\:{X}^{\mathrm{3}} {P}\:−{P}\:'\:\equiv\:\mathrm{1}−{X}^{\mathrm{2}} \:{mod}\left({X}^{\mathrm{4}} +\mathrm{1}\right) \\ $$$$\left.\mathrm{3}\right)\:{P}\:^{\mathrm{2}} −{X}^{\mathrm{3}} {P}−{X}^{\mathrm{2}} \:\:\equiv\:\mathrm{0}\:{mod}\left({X}^{\mathrm{2}} +\mathrm{2}\right) \\ $$

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