Polynomial Ring

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Given a commutative ring with unity R. The polynomial ring R[X].

Characterizing the underlying set of this ring is surprisingly difficult. Start with the set of infinite sequences with elements in R (denoted R^ω) and restrict it to sequences that have a finite non-zero prefix:

R[X] = \set{\p{s_0, s_1, s_2, …} ∈ R^ω : \exists_{n ∈ ℕ}\ \forall_{i > n}\ s_i = 0}

Elements of this set are called polynomials. For a given polynomial p∈R[X] the minimal n ∈ ℕ such that \forall_{i > n}\ p_i = 0 is called the degree of p also denoted \deg p. The zero polynomial is \p{0,0,…} has degree 0 and is the only polynomial where p_{\deg p} = 0.

addition +:R[X]×R[X]→R[X] is defined element-wise using addition from R.

p + q = \p{p_0 + q_0, p_1 + q_1, …, p_i + q_i, …}

multiplication ⋅:R[X]×R[X]→R[X] is defined as a convolution

p⋅q = \p{p_0⋅q_0, p_0⋅q_1 + p_1⋅q_0, …, \sum_{j∈[0,i]} p_{j}⋅q_{i-j}, …}

Under these operations polynomials are a ring.

There is a monomorphism R→R[X] that maps R to constants a ↦ \p{a,0,0,…}.

Scalar multiplication ⋅:R×R[X]→R[X] is defined with the monomorphism as \p{a,p} ↦ \p{a,0,0,…}⋅p.

With this operation polynomials are a vector space.

evaluation \operatorname{apply}:R[X]×R→R

\p{p,a} ↦ \sum_{i∈ℕ} p_i ⋅ a^i

composition ∘:R[X]×R[X]→R[X]

\p{p,q} ↦ \sum_{i∈ℕ} p_i ⋅ a^i

A linear operator is a function f:M → M is an endomorphism of a module.

A derivation on a ring is a linear operator on the underlying module that satisfies

f(a ⋅ b) = a⋅f(b) + f(a)⋅ b .

The derivative \partial: R[X]→R[X] is defined as

\partial p = \p{p_1, 2⋅p_2, …, \p{i+1}⋅p_{i+1},…}

todo: This requires ℕ→R.

All derivations have the form a⋅\partial + b (in operator notation).

modular reduction +:R[X]×R[X]→R[X]