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 $+:R[X]×R[X]→R[X]$