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Intro to Abstract Algebra Fall 2025

Blake Farman

Chapter 15 Rings of Polynomials

Section 15.1 Basic Definition

Definition 15.1. Polynomial.

Assume \(R\) is a ring. A polynomial in the indeterminate \(x\) with coefficients from \(R\) is a formal sum
\begin{equation*} f(x) = \sum_{n=0}^\infty a_nx^n\text{,} \end{equation*}
where for all \(0 \leq n \in \Z\text{,}\) \(a_n \in R\) and for all but finitely many \(0 \leq n \in \Z\text{,}\) \(a_n = 0\text{.}\)
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For each \(0 \leq n \in \Z\text{,}\) the ring element \(a_n\) is called the coefficient of \(x^n\); collectively, we say \(\{a_n\}_{n = 0}^\infty\) are the coefficients of \(f\). The polynomial with all zero coefficients is called the zero polynomial. When the meaning is clear from context, we simply write \(0\) for the zero polynomial.
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Definition 15.2. Degree of a Polynomial.

Assume \(R\) is a ring. For every non-zero polynomial with coefficients from \(R\text{,}\) \(f\text{,}\) the degree of \(f\) is the index of the largest non-zero coefficient. That is to say,
\begin{equation*} \deg{f} = \max\left\{n \in \N \;\middle\vert\; a_n \neq 0\right\}\text{.} \end{equation*}
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We adopt the convention that \(\deg{0} = -\infty\text{.}\)
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Remark 15.3.

Note that every \(a \in R \setminus\{0\}\) can be regarded as a polynomial \(a = \sum_{n=0}^\infty a_nx^n\text{,}\) where
\begin{equation*} a_n = \begin{cases} a \amp\text{if}\ n = 0\\0 \amp\text{else}\end{cases}\text{.} \end{equation*}
This provides a canonical way to identify the ring \(R\) as the subring of \(R[x]\) consisting of elements of degree zero.
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Example 15.4.

Our definition coincides with the familiar definition of polynomial from basic algebra, but allows for coefficients that are not necessarily numbers.
  • The degree two polynomial \(f(x) = \pi x^2 + 3x - 7\) has coefficients from \(\R\) (or \(\C\)).
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  • The degree seven polynomial \(h(x) = 3x^7 - 5x^2 + 8\) has coefficients from \(\Z\) (or \(\Q\) or \(\R\) or \(\C\)).
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  • The degree three polynomial \(h(x) = \overline{7}x^3 + 3\) has coefficients from \(\Z/8\Z\text{.}\)
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Section 15.2 The Ring Structure

Definition 15.5.

Assume \(R\) is a ring. The set of all polynomials in the indeterminate \(x\) with coefficients from \(R\) is denoted \(R[x]\text{.}\)
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Definition 15.6. Polynomial Addition.

Assume \(R\) is a ring. Let
\begin{equation*} f(x) = \sum_{n=0}^\infty a_nx^n \quad\text{and}\quad g(x) = \sum_{n=0}^\infty b_nx^n \end{equation*}
be polynomials with coefficients from \(R\text{.}\) The sum of \(f\) and \(g\) is the polynomial
\begin{equation*} (f+g)(x) = \sum_{n=0}^\infty (a_n + b_n)x^n\text{.} \end{equation*}
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Proof.

Let \(f,g,h \in R[x]\) be given. Write \(a_n, b_n, c_n\) for the coefficients of \(f\text{,}\) \(g\text{,}\) and \(h\text{,}\) respectively.
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Since addition on \(R\) is associative,
\begin{align*} (f + (g + h))(x) \amp= \sum_{n=0}^\infty (a_n + (b_n + c))x^n\\ \amp=\sum_{n=0}^\infty ((a_n + b_n) + c_n)x^n\\ \amp= ((f+g)+h)(x) \end{align*}
Hence polynomial addition is associative.
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The identity element for polynomial addition is the zero polynomial because
\begin{align*} (0 + f)(x) \amp= \sum_{n=0}^\infty (0 + a_n)x^n\\ \amp= \sum_{n=0}^\infty a_nx^n\\ \amp= f(x)\\ \amp= \sum_{n=0}^\infty (a_n + 0)x^n\\ \amp= (f+0)(x)\text{.} \end{align*}
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The additive inverse of \(f\) is the polynomial \(-f\) with coefficients \(-a_n\) because
\begin{align*} (f + -f)(x) \amp= \sum_{n=0}^\infty (a_n - a_n)x^n\\ \amp= \sum_{n=0}^\infty 0x^n\\ \amp= 0\\ \amp= \sum_{n=0}^\infty (-a_n + a_n)x^n\\ \amp= (-f + f)(x) \end{align*}
Therefore \(R[x]\) is an abelian group under polynomial addition.
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Definition 15.8.

Assume \(R\) is a ring. Let
\begin{equation*} f(x) = \sum_{n=0}^\infty a_nx^n \quad\text{and}\quad g(x) = \sum_{n=0}^\infty b_nx^n \end{equation*}
be polynomials with coefficients from \(R\text{.}\) The product of \(f\) and \(g\) is the polynomial \(fg(x) = \sum_{n=0}^\infty c_n\text{,}\) where
\begin{equation*} c_n = \sum_{i=0}^n a_i b_{n-i} = \sum_{i+j = n} a_ib_j\text{.} \end{equation*}
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Example 15.9.

The product of the polynomials \(f(x) = 3x^2 + 4x - 7\) and \(g(x) = 2x^3 - x^2 + x + 3\) is
\begin{align*} fg(x) \amp= \left(-7(2) + 4(-1) + 3(1)\right)x^3\\ \amp\phantom{=}+ \left(-7(-1) + 4(1) + 3(3)\right)x^2\\ \amp\phantom{=}+ \left(-7(1) + 4(3)\right)^x\\ \amp\phantom{=}+ \left(-7(3)\right)\\ \amp= -15x^3 + 20x^2 + 5x -21 \end{align*}
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Proof.

The proof is straightforward, but extraordinarily tedious and provides no interesting information. We omit the details.
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Definition 15.11. Polynomial Ring.

Assume \(R\) is a ring. The polynomial ring in the indeterminate \(x\) with coefficients from \(R\) is denoted by \(R[x]\text{.}\)
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Remark 15.12.

We note that as a direct consequence, we can define the ring of polynomials in the indeterminate \(x\) and \(y\) with coefficients in \(R\) using the convention
\begin{equation*} R[x,y] = \left(R[x]\right)[y]\text{.} \end{equation*}
That is, we view polynomials in the indeterminates \(x\) and \(y\) with coefficients from \(R\) as polynomials in the indeterminate \(y\) with coefficients from the polynomial ring \(R[x]\text{.}\)
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This allows us to recursively define multivariate polynomial rings
\begin{equation*} R[x_1, x_2, \ldots, x_{n+1}] = \left(R[x_1, x_2, \ldots, x_n]\right)[x_{n+1}]\text{.} \end{equation*}
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This formulation makes it easy to define the ring structure on \(R[x_1,\ldots,x_n]\text{.}\) However, in practice, it is not always easy to work with this formulation. One defines the total degree of the monomial \(x_1^{a_1}x_2^{a_2} \cdots x_n^{a_n}\) to be \(a_1 + a_2 + \cdots + a_n\text{.}\) If we take the indexing set
\begin{equation*} I = \left\{(a_1, a_2, \ldots, a_n) \;\middle\vert\; a_i \geq 0\right\} \end{equation*}
then we can use the total degree to provide a representation similar to the one for the polynomial ring with a single indeterminate by writing
\begin{equation*} \sum_{\alpha \in I} c_{\alpha} x^\alpha \end{equation*}
where \(c_\alpha \in R\) and if \(\alpha = (a_1, a_2, \ldots, a_n)\text{,}\) then
\begin{equation*} x^{\alpha} = x_1^{a_1} x_2^{a_2} \cdots x_n^{a_n}\text{.} \end{equation*}
Note that only finitely many \(c_\alpha\) are non-zero.
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Proof.

We leave this as an exercise for the reader.
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Exercises 15.3 Exercises for Undergrads & Grads

1.

Prove that if \(R\) is an integral domain, then so is \(R[x]\text{.}\)
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Hint.
Consider the leading term of the product of two polynomials.
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2.

Assume \(R\) is a commutative ring. Fix \(\alpha \in R\text{.}\) Define the function
\begin{align*} \operatorname{Ev}_\alpha \colon R[x] \amp\to R\\ p \amp\mapsto p(\alpha)\text{,} \end{align*}
where \(p = \sum_{n=0}^\infty a_n x^n\) is a polynomial with coefficients from \(R\) and \(p(\alpha)\) is the result of evaluating the function \(p\) at \(\alpha\text{,}\) \(\sum_{n=0}^\infty a_n \alpha^n \in R\text{.}\) Prove that \(p\) is a surjective morphism of rings.
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