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 Algebra Notes
04-02-2015, 05:11 PM
Post: #4
 elim Moderator     Posts: 581 Joined: Feb 2010 Reputation: 0
1.4 Polynomial rings

1.4.1 Definition A polynomial over a ring $R$ with the indeterminate $t$ is an expression
$a_n t^n +a_{n-1}t^{n-1}+\cdots +a_1 t + a_0 \tag{1}$$\qquad$where $n\in\mathbb{N},\; a_j \in R\,(j = \overline{0,n})$ We call $a_j$ the coefficient of $t^j\;(a_j = 0\;(j > n))$
$\qquad$If $D_f:=\{j\mid a_j\ne 0\}\ne\varnothing$, we say polynomial $f$ in (1) has degree $\deg(f) = \max D_f$
$\qquad$otherwise $\deg(f) := -\infty$. $a_m$ is called the leading coefficient of $f$ if $m = \deg(f)\ge 0$.

$\qquad f$ is said to be monic if its leading coefficient is $1$.

$\qquad$Given two polynomials $\sum a_j t^j,\; \sum b_j t^j,$ we define their sum to be $\sum (a_j + b_j) t^j$.

$\qquad$and their product to be $\quad\small{ \displaystyle{\sum c_j t^j\quad(c_j = \sum_{j = i+k} a_i b_k)}}.$
$\qquad$In this way $\displaystyle{R[t] = \{{\small{\sum_{j=0}^n}} a_j t^j\mid n\in\mathbb{N},\; a_j \in R\}}$ becomes a polynomial ring over
$\qquad R$ in the indeterminate $t$, which contains subring $R$ (the set of degree-0 polynomials)

1.4.2 Definition If $f\in R[t]$ is as in (1) and $\alpha\in R$, then the evaluation of $f$ at $\alpha$ is$f(\alpha) = a_n\alpha^n + a_{n-1}\alpha^{n-1}+\cdots +a_1\alpha + a_0\in R$$\qquad$Any fixed $\alpha\in R$ determines an evaluatin homomorphism $R[t]\to R\;(f\mapsto f(\alpha))$.

1.4.3 Lemma If $R$ is an integral domain, so is $R[t].$

1.4.4 Remark Since $R[t]$ is an integral domain whenever $R$ is, it determines a field of fractions
$\qquad R(t)$ called the field of rational functions over $R$ in the indeterminate $t$, which is expressible
$\qquad$as the quotient of polynomials.

$\qquad$If $\phi:R\to S$ is a ring homomorphism, then $\sum a_j t^j \mapsto \sum \phi(a_j)a^j$ determines an induced map
$\qquad R[t]\to S[t]$ which is clearly a ring homomorphism; we often use the same notation $\phi$ for it.
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 Messages In This Thread Algebra Notes - elim - 04-02-2015, 11:29 AM 1.2 Prime subfields - elim - 04-02-2015, 02:02 PM 1.3 Fields of fractions - elim - 04-02-2015, 04:40 PM 1.4 Polynomial rings - elim - 04-02-2015 05:11 PM 1.5 Polynomial rings over fields - elim - 04-03-2015, 04:40 PM RE: Algebra Notes - elim - 04-04-2015, 08:22 PM Notes - elim - 04-05-2015, 07:51 AM Notes - elim - 04-05-2015, 08:02 AM Notes - elim - 04-06-2015, 07:05 AM 2 Field Extensions - elim - 04-06-2015, 07:24 AM Notes - elim - 04-08-2015, 02:58 PM Notes - elim - 04-08-2015, 02:58 PM

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