Polynomial Rings Over A Field

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Definition (Polynomial Rings Over A Field)

Let $F$ be a field. We denote by $F[x]$ the set of all polynomial in the variable $x$ with coefficients in $F$ together with polynomial addition and multiplication (if needed, see page 294 in your text for a formal description of something you are very familiar with). The set $F[x]$ is called the polynomial ring over $F$ (in the indeterminate $x$). We can write each element $p(x)\in F[x]$ in the form $$p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_2x^2+a_1x+a_0$$ for some $n$ where $a_i\in F$ for each $i$ and $a_n\neq 0$ if $n\geq 1$.

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