Algebra Notes

04022015, 05:11 PM
Post: #4




1.4 Polynomial rings
1.4.1 Definition A polynomial over a ring $R$ with the indeterminate $t$ is an expression \[ a_n t^n +a_{n1}t^{n1}+\cdots +a_1 t + a_0 \tag{1}\]$\qquad$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 degree0 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\[f(\alpha) = a_n\alpha^n + a_{n1}\alpha^{n1}+\cdots +a_1\alpha + a_0\in R \]$\qquad$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. 

« Next Oldest  Next Newest »

Messages In This Thread 
Algebra Notes  elim  04022015, 11:29 AM
1.2 Prime subfields  elim  04022015, 02:02 PM
1.3 Fields of fractions  elim  04022015, 04:40 PM
1.4 Polynomial rings  elim  04022015 05:11 PM
1.5 Polynomial rings over fields  elim  04032015, 04:40 PM
RE: Algebra Notes  elim  04042015, 08:22 PM
2 Field Extensions  elim  04062015, 07:24 AM
