Kummer theory

Let $n\ge 1$ be prime to the characteristic of a field $K$, and suppose $K$ contains all $n$th roots of unity. Kummer theory classifies finite abelian extensions of exponent dividing $n$ by finite subgroups of $K^{\times }/(K^{\times }{)}^n$: to a subgroup generated by classes $a_i$ associate $K(a_i^{1/n}:i)$. The pairing sending an automorphism and a radicand class to $\sigma (a^{1/n})/a^{1/n}$ identifies the Galois group with the character group of the radicand subgroup.

In particular, cyclic extensions of degree $n$ correspond to cyclic subgroups of order $n$, rather than to a unique choice of one generator. The cohomological form isFor $n=2$ in characteristic different from two, the nonzero square classes other than the identity classify quadratic extensions. The assumptions on roots of unity and characteristic ensure that the radicals form separable Galois extensions.^[ref]

The cube-class group of a field $K$ is the quotient abelian groupTwo nonzero elements $a,b$ determine the same class precisely when $a/b$ is a cube in $K$. Multiplication of representatives gives the quotient operation. Every class has order dividing three because the cube of its representative is trivial in the quotient.

For $K=\mathbb{R}$, every nonzero real number has a real cube root, so the group is trivial. For a finite field ${\mathbb{F}}_q$, cyclicity of ${\mathbb{F}}_q^{\times }$ gives a cube-class group of order $\gcd (3,q-1)$. In $\mathbb{Q}$, the class of two is nontrivial and has order three: prime exponents in a rational cube must be divisible by three.

When the characteristic is not three and $K$ contains the cube roots of unity, Kummer theory connects these classes with extensions formed by adjoining cube roots. The quotient group itself is defined without those additional Kummer hypotheses.

For a number field and finite subgroup $\Delta \subset K^{\ast }/K^{\ast n}$, adjoining all the indicated nth roots gives an extension unramified outside the displayed finite set. The valuations modulo $n$ are independent of representatives. At a finite place outside $S$, divide each radicand by a local nth power of a uniformizer to obtain a unit. Its polynomial has separable reduction because $n$ is invertible. A finite unramified extension containing all residue roots contains all lifted roots by Hensel's lemma. This proof does not require ${\mu }_n\subset K$.

For a cyclic Galois extension $M/K$ of degree $m$, choose a generator $g$ and a primitive $m$th root of unity $\zeta \in K$. Reindexing the displayed sum gives $gR(w)=\zeta R(w)$. Linear independence of distinct field embeddings ensures this sum is not identically zero as a function of $w$. Choosing $\alpha =R(w)\ne 0$ therefore gives $m$ distinct conjugates ${\zeta }^j\alpha$, while ${\alpha }^m\in K$. Consequently $M=K(\alpha )$ is the splitting field of $X^m-{\alpha }^m$. Independence follows by taking a shortest supposed linear relation among distinct automorphisms, evaluating at $uv$ and subtracting a suitable multiple of the relation at $v$ to remove one term but not all terms.

For nontrivial even $\chi$ in the full cyclotomic tower of $\mathbb{Q}$, Kummer theory relates the p-ramified Iwasawa module to the reflected odd unramified Iwasawa module, up to pseudo-isomorphism. Here $\iota (\gamma )={\gamma }^{-1}$ and $(1)$ is the Tate twist. The induced substitution on characteristic series is $T\mapsto u(1+T{)}^{-1}-1$, where $u$ is the cyclotomic value of $\gamma$. Omitting either operation changes the interpolation convention.

When the characteristic differs from $p$, the Kummer sequence gives $0\to H^n(L,K_s^{\times })/p\to H^{n+1}(L,{\mu }_p)\to H^{n+1}(L,K_s^{\times })[p]\to 0$. For $n\ge 1$, the positive-degree multiplicative groups are torsion. Thus the middle group is zero exactly when the first multiplicative group is $p$-divisible and the second has zero $p$-primary torsion.

For a number field $K$, a finite set $S$ and $m\ge 2$, let $K(S,m)$ consist of classes $[a]\in K^{\ast }/K^{\ast m}$ whose valuations outside $S$ are divisible by $m$. It fits into $0\to {\mathcal{O}}_{K,S}^{\times }/({\mathcal{O}}_{K,S}^{\times }{)}^m\to K(S,m)\to \mathrm{Cl}({\mathcal{O}}_{K,S})[m]\to 0$. The S-unit group is finitely generated and the ideal class group is finite, so $K(S,m)$ is finite. When $S$ contains primes dividing $m$, this controls unramified multiplicative Kummer classes outside $S$.

The square-class group identifies two nonzero elements of a field when their quotient is a square. Multiplication makes it an abelian group of exponent two, hence a vector space over ${\mathbb{F}}_2$. It records quadratic extensions and the images of Kummer theory for $n=2$.

The valuation contributes one factor $C_2$. The unit decomposition of the 2-adic Gaussian field contributes ${\mu }_4/{\mu }_4^2\cong C_2$ and the additive quotient ${\mathbb{Z}}_2[i]/2{\mathbb{Z}}_2[i]\cong C_2^2$. Thus there are sixteen square classes and fifteen quadratic extensions over this field.

The square class of $a\in K^{\ast }$ is its image in the square-class group of a field $K^{\ast }/K^{\ast 2}$. Two nonzero elements have the same square class exactly when their quotient is a square. Every element of this quotient has order dividing $2$. Over the rational numbers, a square class has a unique signed squarefree integer representative.

For a finite extension $K/{\mathbb{Q}}_p$, its square-class group of a field is $K^{\times }/(K^{\times }{)}^2$. When $p$ is odd, parity of the valuation and the square class of the unit residue give $C_2\times C_2$. Representatives are $1,u,\pi ,u\pi$, where $\pi$ is a uniformizer of $K$ and $u$ has nonsquare residue. The Hensel lemma lifts square roots of units with square residue.

For $K={\mathbb{Q}}_2$, the group is $C_2^3$, with generators $-1,5,2$; a unit is a square exactly when it is one modulo eight. More generally, if $K/{\mathbb{Q}}_2$ has degree $d$, the group has order $2^{d+2}$. Indeed, the principal-unit group of a mixed-characteristic local field has a finite cyclic 2-power torsion factor and a free ${\mathbb{Z}}_2$ factor of rank $d$, each contributing its quotient by squares, and the valuation contributes one further factor $C_2$. The specific representatives $1,u,p,up$ therefore apply to ${\mathbb{Q}}_p$ for odd $p$; a ramified extension needs its own uniformizer.

Suppose $n$ is prime to the characteristic of a field $K$ and $K$ contains all $n$th roots of unity. For $a\in K^{\times }$, adjoining a root of $X^n-a$ gives a Kummer extension. Every conjugate of a chosen root $\alpha$ is $\zeta \alpha$ with $\zeta \in {\mu }_n\subseteq K$, so the extension is a cyclic Galois extension of degree dividing $n$.

Adjoining several such radicals gives the finite abelian extensions of exponent dividing $n$ described by Kummer theory. The characteristic restriction is essential: in characteristic $p$, adjoining a $p$th root of an element that is not a $p$th power produces a purely inseparable extension, with no nonidentity automorphisms. Cyclic extensions of degree $p$ in that characteristic are described instead by Artin–Schreier theory.