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.

• 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 $\mathbb{Q}[x]$.