Polynomial Rings

The polynomial ring $R[x]$ consists of all formal sums $\sum_{i=0}^{n} a_i x^i$ with $a_i \in R$. It is a ring under the usual addition and multiplication of polynomials.

Theorem 10.1 (Division Algorithm). If $F$ is a field and $f, g \in F[x]$ with $g \neq 0$ Then There exist unique $q, r \in F[x]$ such that $f = qg + r$ with $\deg(r) \lt \deg(g)$ or $r = 0$.

Theorem 10.2 (Factor Theorem). $a \in F$ is a root of $f \in F[x]$ if and only if $(x - a)$ divides $f$.

Proposition 10.3. A polynomial of degree $n$ over a field has at most $n$ roots (counting multiplicity).

A non-constant polynomial $f \in F[x]$ is irreducible if it cannot be factored as $f = gh$ With both $g$ and $h$ of degree less than $\deg(f)$.

Proposition 10.4. Every polynomial in $F[x]$ factors uniquely into irreducible polynomials (up to Reordering and multiplication by units).

Theorem 10.5 (Eisenstein’s Criterion). Let $f(x) = a_n x^n + \cdots + a_1 x + a_0 \in \mathbb{Z}[x]$. If there exists a prime $p$ such that:

1. $p$ divides $a_0, a_1, \ldots, a_{n-1}$.
2. $p$ does not divide $a_n$.
3. $p^2$ does not divide $a_0$.

Then $f$ is irreducible in $\mathbb{Q}[x]$.

Proof. Suppose $f = gh$ with $g, h \in \mathbb{Z}[x]$ (by Gauss’s lemma), $\deg(g) = r \geq 1$ $\deg(h) = s \geq 1$, $r + s = n$. Modulo $p$: $\bar{f} = \bar{g}\bar{h} = \bar{a}_n x^n$ in $(\mathbb{Z}/p\mathbb{Z})[x]$. Since $\mathbb{Z}/p\mathbb{Z}[x]$ is an integral domain, $\bar{g} = bx^r$ And $\bar{h} = cx^s$ for some $b, c \in \mathbb{Z}/p\mathbb{Z}$. In particular, the constant terms of $g$ and $h$ are both divisible by $p$. But then $p^2$ divides $a_0$ Contradicting condition (3). $\blacksquare$

Problem. Show that $x^2 + 1$ is irreducible in $\mathbb{R}[x]$ but reducible in $\mathbb{C}[x]$.

Problem. Use the Euclidean algorithm to compute $\gcd(x^4 + x^3 + x^2 + x + 1, x^3 + 1)$ in $\mathbb{Q}[x]$.

Problem. Show that $f(x) = x^5 - 5x + 3$ is irreducible in $\mathbb{Q}[x]$.

• If $F \subseteq K$ is a field extension, then $F[x]$ is a subring of $K[x]$.
• For a field $F$, the polynomial ring $F[x]$ is a Euclidean domain, hence a PID and a UFD.
• If $R$ is a UFD, then $R[x]$ is a UFD (Gauss’s lemma extends unique factorisation).
• If $R$ is a PID, then $R[x]$ is a UFD but generally not a PID.
• The degree map $\deg : F[x] \setminus \{0\} \to \mathbb{N}$ is a Euclidean valuation.

Polynomial rings extend arithmetic to algebraic structures. The ring F[x] consists of formal sums of powers of x with coefficients in a field, and it inherits the familiar operations of addition and multiplication. The division algorithm allows you to divide one polynomial by another, producing a quotient and remainder, just as with integers. This makes polynomial rings Euclidean domains, where unique factorisation into irreducibles holds. Eisenstein’s criterion provides a practical test for irreducibility: if a prime divides all coefficients except the leading one, and its square does not divide the constant term, the polynomial is irreducible. Polynomial rings are fundamental to coding theory, cryptography, and algebraic geometry.

• Assuming that irreducibility in $R[x]$ implies irreducibility in $S[x]$ for an extension $S \supseteq R$. The polynomial $x^2 + 1$ is irreducible over $\mathbb{Q}$ but reducible over $\mathbb{C}$.
• Forgetting that Eisenstein’s criterion requires the leading coefficient not to be divisible by $p$. If $p \mid a_n$ the criterion gives no information about reducibility.
• Confusing the polynomial ring $F[x]$ with the ring of formal power series $F[[x]]$. In $F[[x]]$ every element is a unit except those with zero constant term.
• Neglecting to check that a proposed factorisation respects the coefficient ring. Over $\mathbb{Z}[x]$, a factorisation into monic polynomials in $\mathbb{Q}[x]$ may not lie in $\mathbb{Z}[x]$.

• Coding theory: BCH and Reed-Solomon codes are constructed using polynomial rings over finite fields $\mathbb{F}_q$.
• Cryptography: The ring $\mathbb{Z}_p[x]/(f(x))$ where $f$ is irreducible gives a finite field $\mathbb{F}_{p^n}$, used in elliptic curve cryptography.
• Algebraic geometry: The coordinate ring of an affine variety is a quotient of a polynomial ring.
• Signal processing: The discrete Fourier transform can be viewed as evaluating a polynomial at roots of unity.

Problem. Show that $f(x) = x^4 + 4x + 2$ is irreducible in $\mathbb{Q}[x]$.

• Euclidean Domains, PIDs, and UFDs — The classification of $F[x]$ as a Euclidean domain, PID, and UFD is central to polynomial ring theory.

• Worked Examples — Several worked examples apply irreducibility criteria and division algorithms developed in this chapter.

• Common Pitfalls — The common pitfalls section addresses frequent errors in applying Eisenstein’s criterion and checking irreducibility.

• Classical Mechanics

• Electromagnetism

• Additional Results
• Classification of Groups of Small Order
• Common Pitfalls
• Euclidean Domains, PIDs, and UFDs