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Ring of polynomials

Definition

Let $(r_0, ..., r_n)$ be a sequence in a commutative ring $R$. The polynomial (in indeterminate $t$) with these coefficients is $\sum_{i=0}^n r_i t^i = r_0 + r_1 t + r_2 t^2 + ... + r_n t^n$.

The formal polynomials (in indeterminate $t$) with coefficients in $R$ form a ring, denoted $R[t]$.

These polynomials are not themselves functions, but do each determine a function $R \to R$, where $a$ is mapped to $\sum_{i=1}^n at^i$.

The field of fractions of this ring is called the field of rational expressions and denoted $R(t)$

Formal construction

Addition is defined by $\sum_{i=0}^n r_i t^i + \sum_{i=0}^m s_i t^t = \sum_{i=0}^{\max(m,n)} (r_i + s_i) t^i$, taking $r_i, s_j = 0$ for $i > n, j > m$.

Multiplication is defined by $(\sum_{i=0}^n r_i t^i) \times (\sum_{i=0}^m s_i t^t) = \sum_{i=0}^n (r_i + s_i) t^i$

Properties

By identifying members of $R$ with constant polynomials, $R$ is a subring of $R[t]$.

The quotient ring $R[t] / \langle t \rangle$ is isomorphic to $R$.

The polynomials $\mathbb{F}[t]$over a field $\mathbb{F}$ form a euclidean domain and therefore a principal ideal domain.

The polynomials $R[t]$ over an integral domain $R$ form an integral domain.

all these pages adapted with probably insufficient credit from my university's lecture notes