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Questions tagged [polynomial-congruences]

Questions about congruences where the modulus is a polynomial. For questions concerning congruences between polynomials where the modulus is an integer, use the tag (modular-arithmetic) instead.

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solve $x^2 -4x +13 \equiv 0 \pmod{81}$?

How do I solve $x^2 -4x +13 \equiv o \pmod{81}$ ? I know that this is the same as $x^2 -4x +13 \equiv x^2 + 2x + 1 \equiv (x +1)^2\equiv 0\pmod{3^4}$ but why is $x \equiv -1\pmod{3}$ the only ...
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Is there a solution for $x^6 \equiv 5 \pmod {71}$? [closed]

How can I verify that a solution exists for $$x^6 \equiv 5 \pmod {71}$$
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What is the congruence class of $x^3\mod x^3+x+1$?
I have a given Polynom congruence with a Polynom $x^3+x+1$ ... so the set of the congruence classes is $\{0, 1,x,x+1,x^2,x^2+1,x^2+x,x^2+x+1\}$ But what would look this like? x^3\mod x^3+x+1\equiv ...
I've been looking at these for over an hour and I don't understand how to do them. Any hints would be greatly appreciated. Let $p(x) = x^3 + x + 1$ and $F = Z_3[x]/\langle p(x)\rangle$. Factor \$p(x)...