Separable Extensions and Cyclotomic Fields

The Galois correspondence needs its extensions to have distinct roots, so that a degree-$n$ polynomial contributes $n$ separate roots for automorphisms to permute. Repeated roots are a defect that appears only in positive characteristic. A formal derivative detects it without leaving the base field, and perfect fields never exhibit it. The resulting theory settles the classification of finite fields and the degree of the cyclotomic fields.

Multiple roots and separability

Over a splitting field, $f(x)=(x-{\alpha }_1{)}^{n_1}\cdots (x-{\alpha }_k{)}^{n_k}$ with distinct ${\alpha }_i$; the exponent $n_i$ is the multiplicity of ${\alpha }_i$, which is a simple root when $n_i=1$ and a multiple root otherwise.

Multiplicity does not depend on which splitting field is used, since any two are isomorphic by a map bijective on roots. So $x^2-2$ is separable over $\mathbb{Q}$ with roots $\pm \sqrt{2}$, while $(x^2-2{)}^n$ for $n\ge 2$ is inseparable.

The derivative test

Detecting a multiple root looks like it requires the splitting field, but the formal derivative does it inside $F[x]$.

This is the calculus formula, but purely algebraic: no limits, valid over any field. It obeys the sum and product rules, provable directly from the definition.

The gcd is computed by the Euclidean algorithm in $F[x]$, so separability is decidable over the base field, no factoring required.

In characteristic $0$ this settles the matter. If $p(x)$ is irreducible of degree $n$, then $D_{x}p$ has degree $n-1$, is nonzero, and shares no factor with the irreducible $p$, so $\gcd (p,D_{x}p)=1$.

Inseparability in characteristic p

The characteristic-$0$ argument used $\mathrm{deg}{D}_{x}p=n-1$. In characteristic $p$ the derivative of $x^{pm}$ is $pm{x}^{pm-1}=0$, so the derivative can drop by more than one degree, and $D_{x}p$ can vanish entirely. When $D_{x}p=0$, every exponent of $p$ is a multiple of the characteristic, so $p(x)=p_1(x^p)$ for some $p_1$. This is the only way an irreducible can be inseparable.

The standard inseparable example is $p(x)=x^2-t$ over the rational function field $F={\mathbb{F}}_2(t)$. It is irreducible (Eisenstein at the prime $t$), but $D_{x}p=2x=0$, and over a splitting field $x^2-t=(x-\sqrt{t}{)}^2$ since the cross term $2x\sqrt{t}$ vanishes in characteristic $2$. The single root $\sqrt{t}$ has multiplicity $2$.

Much of positive-characteristic algebra follows from the vanishing of the binomial cross terms.

The map $a\mapsto a^p$ is the Frobenius endomorphism. On a finite field it is injective and therefore surjective, so every element is a $p$th power.

Perfect fields and finite fields

The finite-field argument generalizes: over a perfect field the trick that turned $p(x)=q(x^p)$ into a $p$th power of a polynomial applies, contradicting irreducibility. So inseparability never arises.

This gives the cleanest route to the finite fields themselves.

The containment ${\mathbb{F}}_{p^d}\subseteq {\mathbb{F}}_{p^n}$ holds exactly when $d\mid n$, so the subfields of a finite field mirror the divisors of $n$. Their Galois structure appears in the finite fields lesson.

Addition is coordinatewise in characteristic $2$, where $x+x=0$, and multiplication follows from ${\alpha }^2=\alpha +1$:

Separable and inseparable degree

When an irreducible $p(x)$ over a characteristic-$p$ field is inseparable, peel off powers of $x^p$ until the derivative is nonzero: there is a unique $k$ and a unique irreducible separable $p_{\mathrm{sep}}$ with $p(x)=p_{\mathrm{sep}}(x^{p^k})$.

An extension $K/F$ is separable if the minimal polynomial of every element of $K$ is separable. Over perfect fields every finite extension is separable, so the distinction only matters for function fields and other imperfect fields of positive characteristic.

Cyclotomic polynomials

Return to characteristic $0$ and organize the roots of unity by their order. The $n$th roots of unity form a cyclic group of order $n$; grouping them by which $d\mid n$ they have as their exact order factors $x^n-1$.

Since every $n$th root of unity is primitive of exactly one order $d\mid n$, the linear factors of $x^n-1$ partition by order:

Comparing degrees recovers the identity $n={\sum }_{d\mid n}\phi (d)$. The factorization is a recursion: ${\Phi }_1=x-1$, and dividing $x^n-1$ by the product of the lower ${\Phi }_d$ gives ${\Phi }_n$.

Two facts make ${\Phi }_n$ the minimal polynomial of ${\zeta }_n$.

Since ${\Phi }_n$ is the minimal polynomial of ${\zeta }_n$, the degree of the cyclotomic field is its degree $\phi (n)$.

The degrees factor the fields into recognizable pieces. For $n=8$, $\phi (8)=4$ and $\mathbb{Q}({\zeta }_8)$ contains both $i$ and ${\zeta }_8+{\zeta }_8^{-1}=\sqrt{2}$, so

the biquadratic field. This settles the constructibility of regular polygons: the $n$-gon is constructible when $\frac{1}{2}\phi (n)$ is a power of $2$, which is why the $17$-gon works and the $7$-gon and $9$-gon do not, as the construction lesson promised.

The cyclotomic fields are the first family of extensions whose automorphisms have an explicit description: $\mathrm{Gal}(\mathbb{Q}({\zeta }_n)/\mathbb{Q})\cong (\mathbb{Z}/n\mathbb{Z}{)}^{\times }$, computed in the cyclotomic and abelian extensions lesson. Separability lets the count of automorphisms match the degree, the standing hypothesis behind the Galois correspondence.