Question

Hint. Use Chinese Remainder theorem [Lesture Note, Lecture 19]. Problem 2 (20 pts). Find all roots of the polynomial x³

+x+1 modulo 27 . Write your answer in natural representatives modulo 27.

Answer 1

The question pertains to solving the polynomial equation x³ + x + 1 modulo 27 using the Chinese Remainder Theorem in the field of Mathematics, specifically at the College level.

Step-by-step explanation:

The subject of the question in focus is Mathematics, specifically in the area of number theory that involves the Chinese Remainder Theorem. This theorem is often used to solve polynomial equations modular arithmetic, which can include finding roots. The process involves reducing the problem to a series of simpler congruences and then finding a solution that satisfies all congruences simultaneously. To solve x³+x+1 modulo 27, one would start by factoring 27 into prime powers and apply the theorem, solving the polynomial congruences for each prime power, and finally, reconstructing the natural representatives modulo 27 that satisfy all congruences. However, the question provides information that seems to be unrelated to solving polynomial congruences, which hints at potential typos in the question prompt.