Abstract Algebra – 1.1 Polynomials

Definition 1.1.1 If $R$ is a commutative ring, then a formal power series over $R$ is a sequence of elements $s_i\in R$ for all $i\ge 0$, called the coefficients of $\sigma$:
$$\sigma = (s_0,s_1,s_2,\dots,s_i,\dots).
We can consider $\sigma$ as a function from $\mathbb N$ to $R$. Denote $R[[x]]$ as the set of all formal power series over $R$.

Let $\sigma=(s_0,s_1,s_2,\dots,s_i,\dots)$ and $\tau=(t_0,t_1,t_2,\dots,t_i,\dots)$ be formal power series.

We define two binary operations, addition and multiplication, on $R[[x]]$.
$$\sigma +\tau := (s_0+t_0,s_1+t_1,\dots,s_i+t_i,\dots);\\
\sigma\cdot \tau := (s_0t_0,s_0t_1+s_1t_0,\dots,\sum_{j=0}^is_jt_{i-j},\dots).

Definition 1.1.2 A polynomial over a commutative ring $R$ is a formal power series $\sigma$ over $R$ for which there exists an integer $n\ge 0$ with $\sigma(i)=0,\forall i>n$.

A polynomial has only finitely many nonzero coefficients. The zero polynomial, denoted by $\sigma = 0$, is the sequence $\sigma = (0,0,0,\dots)$.

Definition 1.1.3 If $\sigma=(s_0,s_1,\dots,s_n,0,0,\dots)$ is a nonzero polynomial, then there is $n\ge 0$ with $s_n\ne 0$ and $s_i=0,\forall i>n$. We call $s_n$ the leading coefficient of $\sigma$, we call $n$ the degree of $\sigma$, and we denote the degree by
$$n = \deg(\sigma)
If the leading coefficient $s_n=1$, the $\sigma$ is called monic.

Specifically, the zero polynomial $0$ does not have a degree because it has no nonzero coefficients.

Denote $R[x]$ as the set of all polynomials over $R$. It's trivial that $R[x]\subset R[[x]]$.

Proposition 1.1.1 If $R$ is a commutative ring, then $R[[x]]$ is a commutative ring that contains $R[x]$ and $R'$ as subrings, where $R'=\{(r,0,0,\dots);r\in R\}$.

Lemma 1.1.1 Let $R$ be a commutative ring and let $\sigma,\tau\in R$ be nonzero polynomials.

  • Either $\sigma\tau=0$ or $\deg(\sigma\tau)\le \deg(\sigma)+\deg(\tau)$.
  • If $R$ is a domain, then $\sigma\tau\ne 0$ and $\deg(\sigma\tau)=\deg(\sigma)+\deg(\tau)$.
  • If $R$ is a domain, $\sigma,\tau\ne 0$ and $\tau\mid\sigma$ in $R[x]$, then $\deg(\tau)\le \deg(\sigma)$.
  • If $R$ is a domain, then $R[x]$ is a domain.

Definition 1.1.4 Let $R$ be a commutative ring. The indeterminate $x\in R[x]$ is
After defining indeterminate, we can denote a formal power series as $s_0+s_1x+s_2x^2+\cdots+s_ix^i+\cdots$.

Now we can describe the usual role of $x$ in $f(x)$ as a variable. Each polynomial $f(x)=s_0+s_1x+s_2x^2+\cdots+s_nx^n\in R[x]$ defines a polynomial function
$$f^\flat:R\to R
by evaluation: If $a\in R$, $f^\flat(a)=s_0+s_1a+s_2a^2+\cdots+s_na^n\in R$. It should be realized that polynomial and polynomial function are distinct objects.

Sometimes, we write $f^\flat$ as $f$.

Definition 1.1.5 Let $K$ be a field. The fraction field $\operatorname{Frac}(K[x])$ of $K[x]$, denoted by $K(x)$, is called the field of rational functions over $K$.

Proposition 1.1.2 If $K$ is a field, then the elements of $K(x)$ have the form $f(x)/g(x)$, where $f(x),g(x)\in K[x]$ and $g(x)\ne 0$.

We usually call $R[x]$ the ring of all polynomials over $R$ in one variable, but also there exist polynomials over $R$ in more than one variables.

Definition 1.1.6 Let $R$ be a commutative ring, $R[x_1,x_2,\dots,x_n]$ is the ring of polynomials over $R$ in $n$ variables. When $n\ge 2$,
Moreover, when $K$ is a field, we can describe $\operatorname{Frac}(K[x_1,x_2,\dots,x_n])$ as all rational functions in $n$ variables


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