Kummer theory

When a field $K$ contains the $n$th roots of unity and its characteristic does not divide $n$, Kummer theory identifies cyclic extensions of degree dividing $n$ with suitable classes in $K^{\times }/(K^{\times }{)}^n$. Cohomologically,

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 odd $p$, parity of valuation and the square class of the unit residue give a group $C_2\times C_2$, represented by $1,u,p,pu$ for a unit $u$ with nonsquare residue. For $p=2$, valuation parity and unit residue modulo eight give $C_2^3$, with generators $-1,5,2$. The unit criteria follow from the Hensel lemma, using its strong derivative version for $p=2$.

When a field $K$ contains the relevant roots of unity, adjoining an $n$th root of an element of $K$ gives a Kummer extension. Such extensions describe abelian extensions of exponent dividing $n$.