A polynomial in one variable with coefficients in a ring \(R\) is defined as a map \(P\colon\mathbb{Z}_{\geq 0}\to R\) such that the image of \(n\in\Z_{\geq 0}\) under \(P\) is denoted by \(P_n\) and the set \(\{n\in\mathbb{Z}_{\geq 0}:P_n\neq 0\}\) is finite.
For two polynomials \(P,Q\) their addition, \(P+Q\) is defined as the addition of functions, i.e., \((P+Q)_n=P_n+Q_n\text{.}\) Their product is defined as follows:
The set of all polynomials in one indeterminate with coefficients in a ring \(R\) is denoted by \(R[X]\text{.}\) The set \(R[X]\) is a ring with unity, where the unity is the polynomial \(1\colon\Z_{\geq 0}\to R\) such that \(0\mapsto 1\) and \(n\mapsto 0\) for all \(n\neq 0\text{.}\)
Note that every polynomial \(P\) can be uniquely written as \(P=\sum P_i\cdot X^i\text{.}\) So, from now onwards write \(r_0+r_1X+\cdots+r_nX^n\) for the polynomial that maps \(i\in\Z_{\geq 0}\) to \(r_i\) for \(i=0,1,\ldots,n\) and to \(0\) for \(i \geq n+1\text{.}\)
As usual, for a nonzero polynomial \(P\) the largest integer \(n\) for which the coefficient of \(X^n\) is not zero is called the leading coefficient of \(P\) and \(n\) is called the degree of \(P\text{.}\) A polynomial \(P\) is called a monic polynomial if the leading coefficient is \(1\text{.}\)
Let \(R\) be a ring and let \(P=r_0+r_1X+\cdots+r_nX^n\) be a polynomial in \(R[X]\text{.}\) The polynomial function \(P(\cdot)\colon R\to R\) is defined as follows:
\begin{equation*}
P(x)=r_0+r_1x+\cdots+r_nx^n\quad\text{ for }x\in R.
\end{equation*}
Indeed, let \(R=\{r_1,r_2,\ldots,r_n\}\) be a finite field. The polynomial \(P=(X-r_1)(X-r_2)\cdots (X-r_n)\) is not a zero polynomial, but the polynomial function \(P\colon R\to R\) is a zero function.
Definition10.Ring of polynomials in several variables.
Let \(n\in\N\text{.}\) The ring of polynomials in \(n\) variables with coefficients in a ring \(R\) is denoted by \(R[X_1,\ldots,X_n]\text{.}\) It is defined in a similar way as above Definition 4.
Let \(A\) be a commutative ring with unity and \(P_1,P_2\in A[X]\) be polynomials. If the leading coefficient of \(P_2\) is invertible in \(A\text{,}\) then there exist unique polynomials \(Q,R\in A[X]\) such that \(P_1=P_2Q+R\) and \(\deg R < \deg P_2\text{.}\)
Suppose that \(A=\Q[X]\text{.}\) Note that \(\Q[X,Y]=A[Y]\text{.}\) As \(Y^2-X^3\) is a monic polynomial in \(A[Y]\text{,}\) there exists \(Q,R\in A[Y]\) such that \(f=Q(X,Y)(Y^2-X^3)+R(X,Y)\) with \(R=a(X)Y+b(X)\text{.}\) Therefore, \(a(t^2)t^3+b(t^2)=0\) for all \(t\in\Q\text{.}\) As \(\Q\) is an infinite field, we get that \(a(Z^2)Z^3+b(Z^2)=0\in\Q[Z]\text{.}\) Hence, by separating the odd and the even degree terms, we get that \(a=0=b\text{,}\) i.e., \(R=0\text{.}\) Therefore, \((Y^2-X^3)\) divides \(f\text{.}\)
Let \(A\) be a commutative ring with unity. Show that \(a\in A\) is a root of a polynomial \(P\in A[X]\) if and only if \((X-a)\) divides \(P\text{.}\)
