Let $x^4-16$ be an element of the polynomial ring $E= \mathbb{Z}[x]$ and use the bar notation to denote passage to the quotient ring $\mathbb{Z}[x]/(x^4-16)$. Find a polynomial of degree $\leq 3$ that is congruent to $7x^{13} -11x^9 + 5x^5-2x^3+3$ modulo $x^4-16$. Can anyone help understand how to mod out polynomials.
[Math] Modulo polynomial in ring theory
ring-theory
Best Answer
Note that modulo $x^4-16$, $x^4-16 = 0$, so $x^4 = 16$. Thus you can replace every copy of $x^4$ with 16. For one, you can write $7x^{13} = 7x\cdot(x^4)^3 = 7x\cdot (16)^3$, and similarly with the other terms in your polynomial.
...you get the idea.