Section 22 Rings of Polynomials

Intro

The Ring $R[x]$ contains the polynomials with coefficients in the ring $R$. We will call x an indeterminate (不定元) rather than a variable.

Definition of a polynomial as an infinite formal sum ${\sum }_{i=0}^{\infty }{a}_i{x}^i=a_0+a_{1}x+\cdots +a_n{x}^n+\cdots$ so that we have $0+a_{1}x$ and $0+a_{1}x+0x^2$ regarded as the same polynomial

Tips

The degree of $f(x)$ is the largest i such that $a_i\ne 0$. If all $a_i=0$, then the degree of $f(x)$ is undefined.

• It can be proven that $(R[x])[y]$ is isomorphic to $(R[y])[x]$ So we can write $R[x,y]$ as the ring of polynomials in two indeterminates x and y with coefficients in R.

22.4 The Evaluation Homomorphisms for Field Theory

Define the map ${\varphi }_a:F[x]\to E$ as ${\varphi }_{\alpha }(a_0+a_{1}x+\cdots +a_n{x}^n)=a_0+a_1\alpha +\cdots +a_n{\alpha }^n$ $\varphi$ is a homomorphism because: ${\varphi }_{\alpha }(f(x)+g(x))={\varphi }_{\alpha }(f(x))+{\varphi }_{\alpha }(f(x))$ ${\varphi }_{\alpha }(f(x)g(x))={\varphi }_{\alpha }(f(x)){\varphi }_{\alpha }(f(x))$ If $F$ is a field then $F[x]$ is an integral domain, meaning the polynomial is solvable under factorization.