A-10.5 Basis of a Polynomial Ring

Given an $(n-1)$-degree polynomial ring $\mathbb{Z}[X]∕(X^n+1)$, a basis of the polynomial ring is defined to be a set of polynomials that satisfy the following two requirements:

• Linear Independence: Each polynomial in the basis set cannot be expressed as a linear combination of the other polynomials in the same set
• Spanning the Polynomial Ring: A linear combinations of the polynomials in the basis set can express any polynomials in the polynomial ring

Note that for a $(n-1)$-degree polynomial ring, the number of polynomials that form a basis of the polynomial ring is exactly $n$.